GEAR TOOTH OPTIMIZATION FOR NONSTANDARD …

GEAR TOOTH OPTIMIZATION FOR NONSTANDARD CYLINDIRICAL

INTERNAL AND EXTERNAL AEROSPACE GEAR PAIRS

A THESIS SUBMITTED TO

THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF

MIDDLE EAST TECHNICAL UNIVERSITY

BY

ORHAN NURI KARACA

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR

THE DEGREE OF MASTER OF SCIENCE

IN

MECHANICAL ENGINEERING

SEPTEMBER 2019

Approval of the thesis:



submitted by ORHAN NURI KARACA in partial fulfillment of the requirements

for the degree of Master of Science in Mechanical Engineering Department,

Middle East Technical University by,

Prof. Dr. Halil Kalıpçılar

Dean, Graduate School of Natural and Applied Sciences

Prof. Dr. M. A. Sahir Arıkan

Head of Department, Mechanical Engineering

Prof. Dr. Metin Akkök

Supervisor, Mechanical Engineering, METU

Examining Committee Members:

Prof. Dr. F. Suat Kadıoğlu

Mechanical Engineering, Middle East Technical University

Prof. Dr. Metin Akkök

Mechanical Engineering, METU

Prof. Dr. Orhan Yıldırım


Prof. Dr. Ömer Anlağan

Mechanical Engineering, Bilkent University

Assist. Prof. Dr. Orkun Özşahin


Date: 02.09.2019

iv

I hereby declare that all information in this document has been obtained and

presented in accordance with academic rules and ethical conduct. I also declare

that, as required by these rules and conduct, I have fully cited and referenced all

material and results that are not original to this work.

Name, Surname:

Signature:

Orhan Nuri Karaca

v

ABSTRACT



Karaca, Orhan Nuri

Master of Science, Mechanical Engineering

Supervisor: Prof. Dr. Metin Akkök

September 2019, 162 pages

Aerospace gears are produced by grinding method and can have arbitrary addendum

circle and dedendum circle radii and rounded gear root as opposite to the gears

produced by traditional method such as rack and pinion generation method. The

aerospace gears have also nonstandard module, pressure angle and helix angle. In this

study, helical external and internal aerospace gear pairs are optimized by considering

bending stress, contact stress and scuffing limitations. Pinion and gear number of

teeth, normal module, helix angle, pressure angle, pinion shifting coefficient, center

distance, pinion addendum radius, gear addendum radius, pinion dedendum radius and

gear dedendum radius are taken into consideration as the design parameters which are

to be optimized. Pinion and gear rotational speeds, input power, Young Modulus and

Poisson ratios of the pinion and gear material, Contact and Bending strength of the

gear material are taken as input parameters. All the possible gear pairs are considered

in terms of contact stress, bending stress, scuffing temperature, tiff clearance, top land

thickness, root form radius and tip clearance limitations. AGMA 908-B89 is used for

evaluating the contact stress geometry factor. AGMA 2001-D04 is used for evaluating

the bending and contact stress. The rounded gear root evaluation is conducted for

cylindrical gear pairs. The bending stress geometry factor is implemented for gear

pairs which have rounded gear root. AGMA 925 is used for evaluating the flash

vi

temperature and scuffing evaluations. The analytical method is conducted in

MATLAB. All the obtained results are compared with KISSOFT commercial tool and

it is observed that KISSOFT uses only the transverse plane in the gear root evaluation

and does not consider the backlash effect on the bending stress geometry factor

evaluation. The minimum center distance optimization has a crucial role in aerospace

applications to decrease the weight of an air vehicle. The minimum weight design

solutions are obtained by using the minimum center distance optimization.

Keywords: Gear tooth profile optimization, grinding, rounded gear root, aerospace

gear optimization,

vii

ÖZ

HAVACILIKTA KULLANILAN STANDART DIŞI SİLİNDİRİK İÇ VE DIŞ

DİŞLİ ÇİFTLERİNDE DİŞ OPTİMİZASYONU

Karaca, Orhan Nuri

Yüksek Lisans, Makina Mühendisliği

Tez Danışmanı: Prof. Dr. Metin Akkök

Eylül 2019, 162 sayfa

Havacılık dişlileri taşlama yöntemi ile üretilir ve dişli kol-fener dişli gibi geleneksel

yöntemlerle üretilen dişlilerin aksine herhangi bir istenen diş üstü dairesi yarıçapına

ve diş dibi dairesi yarıçapına sahip olabilir. Bu çalışmada havacılıkta kullanılan helisel

iç ve dış dişli çiftlerinin profil optimizasyonu, eğilme gerilimi, temas gerilimi ve

sürtme aşınması limitasyonları yapılmıştır. Havacılık dişlileri ayrıca standart dışı

modül, basınç açısı ve helis açısına sahiplerdir. Döndüren ve döndürülen dişlilerin diş

sayıları, normal modül, helis açısı, basınç açısı, pinyon profil kaydırma katsayısı, dişli

merkezleri arası mesafe, döndüren dişli diş dibi dairesi yarıçapı, dönen dişli diş dibi

dairesi yarıçapı, döndüren dişli diş üstü dairesi yarıçapı ve döndürülen dişlinin diş üstü

dairesi yarıçapı optimize edilmesi gereken tasarım parametreleri olarak alınmıştır.

İhtimal dahilindeki bütün dişli çiftleri temas gerilimi, eğilme gerilimi, sürtünme

aşınması sıcaklığı, tiff aralığı, diş üstü kalınlığı, diş kökü yarıçapı ve diş üstü aralığı

limitasyonları incelenmiştir. AGMA 908–B89 standardı temas gerilimi geometri

faktörü hesaplamasında kullanılır. AGMA 2001-D04 standardı eğilme ve temas

gerilimi hesabı için kullanılır. Yuvarlatılmış dişli kökü hesabı silindirik dişliler için

yapılır. Eğilme gerilimi geometri faktörü hesabı yuvarlatılmış dişli köküne sahip

silindirik dişliler için yapılır. AGMA 925 standardı sürtünme aşınması sıcaklığı

hesabında kullanılır. Analitik metot MATLAB programında kodlamalar geliştirilerek

viii

düzenlenmiştir. Bütün sonuçlar KISSOFT ticari yazılımı ile karşılaştırılmıştır ve

KISSOFT programının dişli kökü hesabında enine düzlemi kullandığı ve dişli

boşluğunun eğilme gerilimi geometri faktörü hesabı üzerindeki etkisini ihmal ettiği

gözlemlenmiştir. Minimum dişli merkezleri arası mesafe optimizasyonu bir hava

aracının ağırlığını azaltmak için önemli bir role sahiptir. Minimum ağırlık tasarım

çözümleri minimum dişli merkezleri arası mesafe optimizasyonu yapılarak elde edilir.

Anahtar Kelimeler: Dişli dişi optimizasyonu, taşlama, yuvarlatılmış dişli kökü,

havacılık dişli optimizasyonu

ix

To my beautiful country…

x

ACKNOWLEDGEMENTS

I would like to thank my supervisor Prof. Dr. Metin Akkök for his supervision, help

and guidance from the beginning to end of this dissertation.

I want to thank Mahir Gökhan Orak for his endless support and encouragement

throughout this study.

I also want to special thanks to my lovely cousin Emirhan Boyraz for his endless

support and encouraging me throughout my research.

I want to thank my family; Zeki Karaca, Hatice Karaca and Hanife Melike Karaca for

their endless support and encouragement throughout this study.

xi

TABLE OF CONTENTS

ABSTRACT ................................................................................................................. v

ÖZ ........................................................................................................................... vii

ACKNOWLEDGEMENTS ......................................................................................... x

TABLE OF CONTENTS ........................................................................................... xi

LIST OF TABLES ..................................................................................................... xv

LIST OF FIGURES ................................................................................................. xix

LIST OF ABBREVIATIONS ................................................................................ xxiii

LIST OF SYMBOLS ............................................................................................. xxiv

CHAPTERS

1. INTRODUCTION AND LITERATURE SURVEY ............................................ 1

1.1. Introduction ....................................................................................................... 1

1.2. Objective of the Thesis ...................................................................................... 3

1.3. Literature Survey ............................................................................................... 3

2. CYLINDIRICAL EXTERNAL AND INTERNAL GEAR PAIR

OPTIMIZATION PARAMETERS AND OPTIMIZION METHODOLOGY............ 9

2.1. Cylindrical External Gear Pair Optimization Parameters ................................. 9

2.1.1. Geometrical Constraints ............................................................................. 9

2.1.1.1. Root Clearance ................................................................................... 10

2.1.1.2. Top Land Thickness Evaluation of External Gear Pairs .................... 12

2.1.1.3. Top Land Thickness Evaluation of Internal Gear Pairs ..................... 16

2.1.1.4. Contact Ratio Evaluation of External Gear Pairs ............................... 19

2.1.1.5. Contact Ratio Evaluation of Internal Gear Pairs ................................ 20

xii

2.1.1.6. Gear Root Evaluation of External Gear Pairs .................................... 22

2.1.1.7. Gear Root Evaluation of Internal Gear Pairs ..................................... 25

2.1.1.8. Involute Clearance ............................................................................. 27

2.1.1.9. Tiff Clearance .................................................................................... 28

2.1.2. Material Based Constraints ...................................................................... 28

2.1.2.1. Contact Stress Number Evaluation of External Gear Pairs ............... 29

2.1.2.2. Bending Strength Geometry Factor and Bending Stress Number

Evaluation ....................................................................................................... 32

Virtual Spur Gear Evaluation ...................................................... 34

Load Angle and Load Radius for External Gear Pairs ................ 35

Virtual Gear Root Evaluation for External Gear Pairs ............... 37

Critical Section Determination for External Gear Pairs.............. 39

Evaluation of the Critical Radius for External Gear Pairs .......... 40

Load Angle and Load Radius for Internal Gear Pairs ................. 42

Virtual Gear Root Evaluation for Internal Gear Pairs ................. 45

Critical Section Determination for Internal Gear Pairs ............... 46

Evaluation of the Critical Radius for Internal Gear Pairs ........... 47

2.1.2.3. Scuffing Evaluation ........................................................................... 49

Profile Radii of Curvature ........................................................... 51

Gear Tooth Velocities and Loads................................................ 53

Load Sharing Factor .................................................................... 54

Evaluation of Maximum Flash Temperature .............................. 55

3. Optimization Methodology ................................................................................ 57

3.1. Root Clearance Criteria ................................................................................... 57

xiii

3.2. Top Land Thickness Criteria ........................................................................... 58

3.3. Contact Stress Criteria ..................................................................................... 59

3.4. Bending Stress Criteria .................................................................................... 63

3.5. Optimization Steps .......................................................................................... 65

3.5.1. Cross Combination of Normal Module, Helix Angle, Pinion Profile

Shifting Coefficient, Pressure Angle and Pinion Number of Teeth ................... 65

3.5.2. Evaluation of Pinion Addendum Radius and Gear Dedendum Radius .... 66

3.5.3. Gear Root Clearance Elimination ............................................................. 70

3.5.4. Pinion Top Land Thickness Elimination .................................................. 71

3.5.5. Evaluation of Gear Addendum Radius and Pinion Dedendum Radius .... 71

3.5.6. Pinion Root Clearance Elimination .......................................................... 74

3.5.7. Gear Top Land Thickness Elimination ..................................................... 74

3.5.8. Contact Ratio Elimination ........................................................................ 75

3.5.9. Involute Clearance Elimination ................................................................ 75

3.5.10. Tiff Clearance Elimination ..................................................................... 76

3.5.11. Contact Stress Elimination ..................................................................... 77

3.5.12. Bending Stress Elimination .................................................................... 77

3.5.13. Scuffing Elimination ............................................................................... 78

4. SENSITIVITY ANALYSIS AND CASE STUDIES ......................................... 79

4.1. Sensitivity Analysis and Case Studies of External Gear Pair and Internal Gear

Pair .......................................................................................................................... 79

4.1.1. Sensitivity Analysis of External Gear Pair ............................................... 79

4.1.2. Addendum and Dedendum Radii Sensitivity of External Gear Pairs ....... 95

4.1.3. Detailed Optimization of Case 1 ............................................................... 98

xiv

4.1.4. Detailed Optimization of Case 6 ............................................................ 100

4.2. Sensitivity Analysis and Case Studies of Internal Gear Pair ........................ 102

4.2.1. Addendum and Dedendum Radii Sensitivity of Internal Gear Pairs ...... 117

4.2.2. Detailed optimization of Case 11 ........................................................... 120

5. VERIFICATION AND DISCUSSION OF THE RESULTS ........................... 123

5.1. Verification of External and Internal Gear Pairs by Using Kissoft .............. 123

5.2. Discussion of the Sudden Changes in the Value of the Objective Function . 127

5.3. Verification of the Bending Stress Geometry Factor Evaluation ................. 139

6. CONCLUSIONS AND FUTURE WORK ....................................................... 141

REFERENCES ........................................................................................................ 145

7. APPENDICES .................................................................................................. 149

xv

LIST OF TABLES

TABLES

Table 1.1 Summary of some studies ........................................................................... 7

Table 2.1 Basic geometric relations .......................................................................... 12

Table 2.2 Maximum Face width Ratio [15] .............................................................. 30

Table 2.3 Basic geometric relations of virtual gear ................................................... 34

Table 2.4 Critical radius algorithm for the pinion ..................................................... 40

Table 2.5 Algorithm of bending strength geometry factor for the pinion ................. 41

Table 2.6 Scuffing Risk [16] ..................................................................................... 49

Table 3.1 Addendum and dedendum coefficient of standard cutters ........................ 57

Table 3.2 Allowable contact stress number, 𝑠𝑎𝑐 , for steel gears [14] ..................... 60

Table 3.3 Stress cycle factor equations for regimes I, II and III [14]........................ 60

Table 3.4 Reliability factors, 𝐾𝑅 [14] ....................................................................... 62

Table 3.5 Allowable bending stress number, 𝑠𝑎𝑡 , for steel gears [14] .................... 63

Table 3.6 Minimum and maximum values of the constraint functions used in

optimization ............................................................................................................... 65

Table 3.7 Initial assignments of the design variables ................................................ 65

Table 3.8 First cross combination of the design variables ........................................ 66

Table 3.9 Evaluation of addendum radius of the pinion ........................................... 66

Table 3.10 Evaluation of dedendum radius of the gear ............................................. 67

Table 3.11 Gear root clearance elimination algorithm .............................................. 70

Table 3.12 Pinion top land thickness elimination algorithm ..................................... 71

Table 3.13 Evaluation of addendum radius of the gear ............................................. 71

Table 3.14 Evaluation of dedendum radius of the pinion ......................................... 72

Table 3.15 Pinion root clearance elimination algorithm ........................................... 74

Table 3.16 Gear top land thickness elimination algorithm ....................................... 75

Table 3.17 Contact ratio elimination algorithm ........................................................ 75

Table 3.18 Pinion involute clearance elimination algorithm..................................... 75

Table 3.19 Gear involute clearance elimination algorithm ....................................... 76

xvi

Table 3.20 Pinion tiff clearance elimination algorithm ............................................ 76

Table 3.21 Gear tiff clearance elimination algorithm ............................................... 76

Table 3.22 Contact stress elimination algorithm ...................................................... 77

Table 3.23 Pinion bending stress elimination algorithm........................................... 77

Table 3.24 Gear bending stress elimination algorithm ............................................. 77

Table 3.25 Scuffing elimination algorithm ............................................................... 78

Table 4.1 Speed and power values for the analysis from Case 1 to Case 5 .............. 79

Table 4.2 Lower and upper bounds and increment values for addendum and dedendum

radii for the analysis from Case 1 to Case 5 .............................................................. 80

Table 4.3 Input parameters of the studies from Case1 to Case 5 .............................. 81

Table 4.4 Optimized design variables (Case1 – Case 5) ........................................... 82

Table 4.5 Speed and power values for the analysis from Case 6 to Case 10 ............ 83

Table 4.6 Lower and upper bounds and increment values for addendum and dedendum

radii for the analysis form Case 6 to Case 10 ............................................................ 83

Table 4.7 Input parameters of the studies from Case 6 to Case 10 ........................... 85

Table 4.8 Optimized design variables (Case6 – Case10) .......................................... 86

Table 4.9 Input parameters and initial assignment of the design variables of Case EX1,

EX2 and EX3 ............................................................................................................. 95

Table 4.10 Optimized design variables (Case EX1 – Case Ex3) .............................. 96

Table 4.11 Input parameters and initial assignment of the design variables of Case

EX4, EX5 and EX6 .................................................................................................... 97

Table 4.12 Optimized design variables (Case EX4 – Case Ex6) .............................. 98

Table 4.13 Input parameters and initial assignment of the design variables of Case 1

Detailed A and Case 1 Detailed B ............................................................................. 99

Table 4.14 Optimized design variables (Case 1 Detailed A – Case 1 Detailed B) . 100


Detailed A and Case 6 Detailed B ........................................................................... 101

Table 4.16 Optimized design variables (Case 6 Detailed A – Case 6 Detailed B) . 102

Table 4.17 Speed and power values for the analysis from Case 11 to Case 15 ...... 102

xvii

Table 4.18 Lower and upper bounds and increment values for addendum and

dedendum radii for the analysis from Case 11 to Case 15 ....................................... 103

Table 4.19 Input parameters of the studies from Case 11 to Case 15 ..................... 104

Table 4.20 Optimized design variables (Case 11- Case15 ) .................................... 105

Table 4.21 Speed and power values for the analysis from Case 16 to Case 20 ...... 106


dedendum radii for the analysis from Case 16 to Case 20 ....................................... 106

Table 4.23 Input parameters of the studies from (Case 16 – Case 20) .................... 108

Table 4.24 Optimized design variables (Case 16 – Case 20) .................................. 109


INT1, INT2 and INT3 .............................................................................................. 117

Table 4.26 Optimized design variables (Case INT1 – CASE INT3) ...................... 118


INT4, INT5 and INT6 .............................................................................................. 119

Table 4.28 Optimized design variables (Case INT4 – CASE INT6) ...................... 120


Detailed A and Case 11 Detailed B.......................................................................... 121

Table 4.30 Optimized design variables (Case 11 Detailed A – Case 11 Detailed B)

.................................................................................................................................. 122

Table 5.1 Results of the verification case study for external gear pairs .................. 124

Table 5.2 Results of the verification case study for internal gear pairs .................. 126

Table 5.3 Initial assignments of case 21 .................................................................. 128

Table 5.4 Center distance variation with normal module, pinion number of teeth and

gear number of teeth in Case 21 ............................................................................... 129

Table 5.5 The maximum addendum radii of the pinion and the gear with normal

module, pinion number of teeth and gear number of teeth in Case 21 .................... 130

Table 5.6 The maximum dedendum radii of the pinion and the gear with normal

module, pinion number of teeth and gear number of teeth in Case 21 .................... 130

Table 5.7 Comparison of the bending stress geometry factor of the optimized gear

pair obtained by Case 1 ............................................................................................ 140

xviii

Table 6.1 Gear root and bending stress geometry factor evaluation for external gear

pairs with 0.250 𝑚𝑚 normal circular backlash ....................................................... 144

Table 6.2 Gear root and bending stress geometry factor evaluation for internal gear

pairs with 0.250 𝑚𝑚 normal circular backlash ....................................................... 144

xix

LIST OF FIGURES

FIGURES

Figure 2-1 Geometrical parameters of internal gear ................................................... 9

Figure 2-2 Geometrical parameters of external gear ................................................. 10

Figure 2-3 Root clearances 𝑐1 and 𝑐2 of an external gear pair [11] ......................... 11

Figure 2-4 Root clearances 𝑐1 and 𝑐2 of an internal gear pair ................................. 11

Figure 2-5 Tooth thickness of external gear pair pinion at reference and operating

circles ......................................................................................................................... 14

Figure 2-6 Top land thickness evaluation of external gear pair pinion ..................... 15

Figure 2-7 Tooth thickness of internal gear at reference and operating circles ........ 17

Figure 2-8 Top land thickness evaluation of internal gear pair................................. 18

Figure 2-9 Transverse plane view of the line of action of an external gear pair ....... 19

Figure 2-10 Transverse plane view of the line of action of an internal gear pair ..... 21

Figure 2-11 Transverse plane view of external gear pair pinion tooth ..................... 23

Figure 2-12 Normal plane view of external gear pair pinion tooth ........................... 24

Figure 2-13 Transverse plane view of internal gear pair gear tooth ......................... 26

Figure 2-14 Normal plane view of internal gear pair gear tooth ............................... 27

Figure 2-15 Pressure angle where external gear tooth comes to point...................... 36

Figure 2-16 Load angle and load radius of an external gear ..................................... 37

Figure 2-17 Normal plane view of virtual spur gear ................................................. 38

Figure 2-18 Critical section evaluation of virtual spur gear...................................... 40

Figure 2-19 Pressure angle where internal gear tooth comes to point ..................... 43

Figure 2-20 Load angle and load radius of an internal gear ..................................... 44

Figure 2-21 Normal plane view of virtual spur gear ................................................. 46

Figure 2-22 Critical section evaluation of virtual spur gear...................................... 47

Figure 2-23 Transverse relative radius of curvature for external gears [16] ............. 51

Figure 2-24 Transverse relative radius of curvature for internal gears ..................... 52

Figure 2-25 Load sharing factor – unmodified profiles [16] .................................... 54

Figure 3-1 Minimum effective case depth for carburized gears, ℎ𝑒 𝑚𝑖𝑛 [14] ......... 58

xx

Figure 3-2 Hardness ratio factor, 𝐶𝐻 (through hardened) [14] ................................ 61

Figure 3-3 Bending strength stress cycle factor, 𝑌𝑁 [14] ........................................ 64

Figure 4-1 Module sensitivity and center distance relation of external gear pair at

relatively higher pinion rotational speed (4000 rpm) ................................................ 88


relatively lower pinion rotational speed (1500 rpm) ................................................. 88

Figure 4-3 Helix angle sensitivity and center distance relation of external gear pair at

relatively higher pinion rotational speed (4000 rpm) ................................................ 90

Figure 4-4 Helix angle sensitivity and center distance relation of external gear pair at


Figure 4-5 Pinion profile shifting coefficient sensitivity and center distance relation

of external gear pair at relatively higher pinion rotational speed (4000 rpm) ........... 92


of external gear pair at relatively higher pinion rotational speed (1500 rpm) ........... 92

Figure 4-7 Pressure angle sensitivity and center distance relation of external gear pair

at relatively higher pinion rotational speed (4000 rpm) ............................................ 94

Figure 4-8 Pressure angle sensitivity and center distance relation of external gear pair

at relatively lower pinion rotational speed (1500 rpm) ............................................ 94

Figure 4-9 Module sensitivity and center distance relation of internal gear pair at



relatively higher pinion rotational speed (3000 rpm) .............................................. 111

Figure 4-11 Helix angle sensitivity and center distance relation of internal gear pair

at relatively lower pinion rotational speed (900 rpm) ............................................. 113


at relatively higher pinion rotational speed (3000 rpm) .......................................... 113


of internal gear pair at relatively lower pinion rotational speed (900 rpm) ............. 114

Figure 4-14 Pressure angle sensitivity and center distance relation of internal gear pair

at relatively lower pinion rotational speed (900 rpm) ............................................. 116

xxi

Figure 4-15 Pressure angle sensitivity and center distance relation of internal gear pair

at relatively higher pinion rotational speed (3000 rpm) ........................................... 116

Figure 5-1 Tooth contact temperature variation of the optimized gear pair in Case 6

with angle of rotation ............................................................................................... 125


with angle of rotation ............................................................................................... 127

Figure 5-3 Variation of safety factor and top land thickness vs pressure angle with -

0.2 profile shifting coefficient (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 = 35 , 𝑧2 = 138 , 𝛽 =

23° , 𝑟𝑎1 = 83.291 𝑚𝑚 , 𝑟𝑎2 = 312.676 𝑚𝑚 , 𝑟𝑓1 = 73.047 𝑚𝑚 , 𝑟𝑓2 =

302.432 𝑚𝑚) ........................................................................................................... 131

Figure 5-4 Variation of safety factor and top land thickness vs pressure angle with 0

profile shifting coefficient (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 = 35 , 𝑧2 = 138 , 𝛽 = 23° , 𝑥1 =

0 , 𝑟𝑎1 = 83.291 𝑚𝑚 , 𝑟𝑎2 = 312.676 𝑚𝑚 , 𝑟𝑓1 = 73.047 𝑚𝑚 , 𝑟𝑓2 =

302.432 𝑚𝑚) ........................................................................................................... 132

Figure 5-5 Variation of safety factor and top land thickness vs pressure angle with -

0.2 profile shifting coefficient (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 = 37 , 𝑧2 = 145 , 𝛽 = 23°, 𝑟𝑎1 =

87.745 𝑚𝑚 , 𝑟𝑎2 = 328.266 𝑚𝑚 , 𝑟𝑓1 = 77.501 𝑚𝑚 , 𝑟𝑓2 = 318.021 𝑚𝑚 ..... 133

Figure 5-6 Variation of safety factor and top land thickness vs pressure angle with


23° , 𝑟𝑎1 = 87.745 𝑚𝑚 , 𝑟𝑎2 = 328.266 𝑚𝑚 , 𝑟𝑓1 = 77.501 𝑚𝑚 , 𝑟𝑓2 =

318.021 𝑚𝑚 ............................................................................................................ 134

Figure 5-7 Variation of safety factor, top land thickness and pinion root clearance vs

pressure angle with -0.2 profile shifting coefficient (𝑚𝑛 = 4.4 𝑚𝑚 , 𝑧1 = 35 , 𝑧2 =

138 , 𝛽 = 23° , 𝑟𝑎1 = 89.386 𝑚𝑚 , 𝑟𝑎2 = 335.555 𝑚𝑚 , 𝑟𝑓1 =

78.392 𝑚𝑚 , 𝑟𝑓2 = 324.561 𝑚𝑚........................................................................... 135

Figure 5-8 Variation of safety factor and top land thickness (𝑚𝑛 = 4.4 𝑚𝑚 , 𝑧1 =

35 , 𝑧2 = 138 , 𝛽 = 23° , 𝑥1 = 0 , 0.2 , 𝑟𝑎1 = 𝑎𝑙𝑙 , 𝑟𝑎2 = 𝑎𝑙𝑙 , 𝑟𝑓1 = 𝑎𝑙𝑙 , 𝑟𝑓2 = 𝑎𝑙𝑙

.................................................................................................................................. 136

xxii


pressure angle with -0.2 profile shifting coefficient (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 = 38 , 𝑧2 =

149 , 𝛽 = 23° , 𝑥1 = 0 , 𝑟𝑎1 = 89.972 𝑚𝑚 , 𝑟𝑎2 = 337.174 𝑚𝑚 , 𝑟𝑓1 =

79.728 𝑚𝑚 , 𝑟𝑓2 = 326.929 𝑚𝑚 .......................................................................... 137

Figure 5-10 Variation of top land thickness and root clearance vs pressure angle with


23° , 𝑟𝑎1 = 89.972 𝑚𝑚 , 𝑟𝑎2 = 337.174 𝑚𝑚 , 𝑟𝑓1 = 79.728 𝑚𝑚 , 𝑟𝑓2 =

326.929 𝑚𝑚 ............................................................................................................ 138

Figure 5-11 Variation of safety factor and top land thickness (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 =

38 , 𝑧2 = 149 , 𝛽 = 23° , 𝑥1 = 0, 0.2 , 𝑟𝑎1 = 𝑎𝑙𝑙 , 𝑟𝑎2 = 𝑎𝑙𝑙 , 𝑟𝑓1 = 𝑎𝑙𝑙 , 𝑟𝑓2 = 𝑎𝑙𝑙

................................................................................................................................. 139

xxiii

LIST OF ABBREVIATIONS

ABBREVIATIONS

𝑆𝐴𝑃 start of active profile

𝐿𝑃𝑆𝑇𝐶 lowest point of single tooth contact

𝐻𝑃𝑆𝑇𝐶 highest point of single tooth contact

𝐸𝐴𝑃 end of active profile

𝐿𝐴𝐶𝑅 low axial contact ratio

xxiv

LIST OF SYMBOLS

𝑎𝑟𝑒𝑓 reference center distance

𝑎𝑤 operating center distance

𝐴 radius of the gear root

𝑏 vertical distance of the center of the circular root

𝑏𝐻(𝑖) semi-width of Hertzian contact band

𝐵𝑀 thermal contact coefficient

𝑐 root clearance

𝐶𝑒 mesh alignment correction factor

𝐶𝑓 surface condition factor

𝐶ℎ helical factor

𝐶𝐻 hardness ratio factor

𝐶𝑚𝑎 mesh alignment factor

𝐶𝑚𝑐 lead correction factor

𝐶𝑚𝑓 face load distribution factor

𝐶𝑝 elastic coefficient

𝐶𝑝𝑓 pinion proportion factor

𝐶𝑅𝑎𝑣𝑔𝑥 surface roughness constant

𝐶𝑝𝑚 pinion proportion modifier

𝐶1, 𝐶2, ……… , 𝐶6 distances along line of action

xxv

𝐶𝜓 helical overlap factor

𝑑 diameter

𝐸 modulus of elasticity

𝐸𝑟 reduced modulus of elasticity

𝐹 effective face width

(𝐹𝑡)𝑛𝑜𝑚 nominal tangential load

𝐹𝑤𝑛 normal operating load

ℎ𝑐(𝑖) central film thickness at a given point

ℎ𝑓 height of Lewis parabola

𝐻𝑐(𝑖) dimensionless central film thickness

𝐼 geometry factor for contact strength

𝐼𝑐 involute clearance

𝑗 circular backlash

𝑘𝑠𝑢𝑚𝑝 parameter for calculating 𝜃𝑚

𝐾 flash temperature constant

𝐾𝐵 rim thickness factor

𝐾𝐷 combined derating factor

𝐾𝑓 stress correction factor

𝐾𝑚 load distribution factor

𝐾𝑜 overload factor

𝐾𝑅 reliability factor

xxvi

𝐾𝑠 size factor

𝐾𝑇 temperature factor

𝐾𝑣 dynamic factor

𝐾𝜓 helix angle factor

𝐿ℎ𝑣 load height

𝐿𝑚𝑖𝑛 minimum length of contact lines

𝐿𝑥 filter cutoff of wavelength x

𝑚 module

𝑚𝐹 axial contact ratio

𝑚𝐺 gear ratio

𝑚𝑁 load sharing ratio

𝑚𝑝 transverse contact ratio

𝑛 rotational speed

𝑝 circular pitch

𝑃𝑏 base pitch

𝑃𝑑 transverse diametral pitch

𝑃𝑛𝑑 normal diametral pitch

𝑃𝑥 axial pitch

𝑄 tail area of the normal probability function

𝑟 reference radius or pitch circle radius

𝑟𝑎 addendum circle radius

xxvii

𝑟𝑏 base circle radius

𝑟𝑓 root radius or dedendum circle radius

𝑟𝑐𝑟 critical radius

𝑟𝑛𝐿 load radius

𝑟𝑡𝑓 tip form radius

𝑅𝑐 reserve factor for contact stress

𝑅𝑓 root form radius

𝑅𝑚 mean radius

𝑅𝑡 reserve factor for bending stress

𝑅𝑎1𝑥, 𝑅𝑎2𝑥

average surface roughness (pinion, gear) at 𝐿𝑥

𝑠 tooth thickness

𝑠𝑐 contact stress number

𝑠𝑡 bending stress number

𝑠𝑏 bending stress number

𝑆𝐹 tooth thickness at critical section

𝑆𝑛 normal tooth thickness in reference circle

𝑆𝑡 transverse tooth thickness in reference circle

𝑠𝑎𝑐 allowable contact stress number

𝑠𝑎𝑡 allowable bending stress number

𝑠𝑛𝑎 top land thickness

𝑠𝑎𝑐𝑝 permissible allowable contact stress number

xxviii

𝑠𝑎𝑡𝑝 permissible allowable bending stress number

𝑡𝑐 tip chamfer

𝑇𝑖𝑓𝑓 tiff clearance

𝑈(𝑖) speed parameter

𝑣𝑒(𝑖) entraining velocity

𝑣𝑠(𝑖) sliding velocity

𝑣𝑟(𝑖) rolling (tangential) velocity

𝑤 rotational velocity

𝑤𝑛 normal unit load

𝑊(𝑖) load parameter

𝑊𝑡 transmitted tangential load

𝑥 profile shift coefficient

𝑦 profile shift

𝑌 tooth form factor

𝑌𝑁 bending strength stress cycle factor

𝑧 number of teeth

𝑍 active length of line of action

𝑍𝑛 contact strength stress cycle factor

𝛼 pressure angle

𝛼𝑛𝐿 load angle

𝛼𝑛𝑊 pressure angle at load application point

xxix

𝛽 reference helix angle

𝛽𝑏 base helix angle

𝜂40 dynamic viscosity at temperature 40°𝐶

𝜂100 dynamic viscosity at temperature 100°𝐶

𝜂𝑚 dynamic viscosity at gear tooth temperature 𝜃𝑚

𝜃𝑚 tooth temperature

𝜃𝑓𝑙(𝑖) flash temperature

𝜃𝑜𝑖𝑙 oil inlet or sump temperature

𝜃𝑠 mean scuffing temperature

𝜆𝑚 heat conductivity

𝜇𝑚(𝑖) mean coefficient of friction

𝜈 Poisson’s ratio

𝜉(𝑖) pinion roll angle at point 𝑖 along the line of action

𝜌 radius of curvature of profiles at point of contact stress

calculation

𝜌𝑚 radius of curvature of the pinion profile at mean radius

𝑋Γ load sharing factor

± upper sign for external pair lower sign for internal pair

∓ upper sign for external pair lower sign for internal pair

&& and operator in MATLAB

1

CHAPTER 1

1. INTRODUCTION AND LITERATURE SURVEY

1.1. Introduction

Gears are widely used in transmission systems to transmit power. There are numerous

gear types used in industrial applications. Cylindrical gears are not only used in

automotive industry, they are also used in aerospace applications especially in

planetary stage and accessory gear box. They are also used as collector gear in some

helicopters. Cylindrical gears used in automotive industry are usually produced by

traditional techniques such as hobbing and shaping. However, aerospace cylindrical

gears are mostly produced by grinding. Grinding manufacturing is also called as direct

gear design in the literature. Aerospace gears can have arbitrary addendum and

dedendum radii since the grinding wheel can be dressed into whatever shape is

desired. Dressing the grinding wheel in desired shape also makes obtaining the

rounded gear root possible. However, the addendum circle and dedendum circle radii

depends on the cutter shifting in traditional manufacturing techniques and obtaining

the arbitrary addendum circle and dedendum circle radii is not possible. This

dependency does not allow the design nonstandard gears.

Numerous different gears can be manufactured by using direct gear design technique.

Therefore, different gear pairs can be manufactured to satisfy a wide range of working

conditions.

Optimization is a required process at the beginning of a design project to obtain the

optimum design which satisfies the requirements and the desired stress values to

maintain a robust design against the compelling operational conditions. Gear design

optimization takes an important place in helicopter transmission system projects. The

minimum weight consideration is one of the most important design approach in

2

aerospace applications. Therefore, the gear pairs which have minimum center distance

are more desirable when the whole system is considered. Minimum center distance

makes the casing weight minimum. Hence, the weight of the gear box is also

minimized.

There are numerous optimization techniques. Traditional optimization techniques are

not suitable for the complex gear optimization. Nontraditional optimization methods

are commonly used in gear optimization due to this complexity. However, the gear

optimization studies conducted by nontraditional optimization method do not cover so

many design parameters as considered in this study. Most of the studies are suitable

to use nontraditional optimization methods due to including less design parameters

and constraint functions as compared to current study. The different optimization

method is required for the current study due to its complexity. The fine sizing

optimization methodology which is explained in a detailed way in the following

sections is determined as the optimization procedure. The fine sizing optimization

method is used in commercial gear design and analysis computer programs like

MASTA and KISSOFT. In this method, all the possible gear pairs are constituted by

giving a range for all design parameters and then by combining all of them. The related

constraints are evaluated for all the possible gear pairs. Gear pairs which do not satisfy

the required constraints are eliminated. The remaining gear pairs after the all constraint

eliminations are completed are the gear pairs which satisfy the all the geometrical and

material strength requirements. Fine sizing method vanishes the possibility of the

converging a local minimum such a complex gear optimization like the current study.

Commercial gear design and analysis computer programs makes the gear optimization

by considering only the gears produced by traditional manufacturing techniques. The

arbitrary addendum radius, dedendum radius and the rounded gear root are not

considered in these programs. Addendum and dedendum radii are dependent on the

shifting coefficient and taking the arbitrary values for these radii is not allowed.

MASTA and KISSOFT programs are also not capable to calculate the bending stress

for the gears which have rounded root. Bending stress geometry factor is calculated as

3

if the gear root is produced by traditional techniques and has a trochoidal shape despite

the software allows the user to implement arbitrary addendum and dedendum radii as

input parameters.

In the current study, the optimum gear pair is obtained by considering the direct gear

design methodology. The rounded gear root and arbitrary addendum and dedendum

radii are considered. Pinion and gear number of teeth, normal module, pressure angle,

helix angle, center distance, shifting coefficient, addendum and dedendum radii of the

pinion and the gear are taken as the design variables. Contact stress, bending stress

and scuffing limitations are considered with the geometrical constraints.

1.2. Objective of the Thesis

The main aim of the study is to obtain optimum helical external and internal gear pairs

for given input and given output speeds under a specified input power. Geometrical

calculations of the helical external and internal gear pairs are evaluated. Bending

stress, contact stress and scuffing calculations are conducted. The optimum gear

design is obtained by using fine sizing method.

1.3. Literature Survey

The conducted studies in the literature about the cylindrical gear pairs are investigated

in this part of the thesis.

Mohan and Seshaiah [1] conducted an optimization study on spur gear sets by taking

the center distance, weight and tooth deflections as objective functions. Module, face

width and number of teeth on pinion were taken into consideration as decision

variables. Since the optimum problem is multi objective function and involves

constraints, it is a very challenging issue to obtain the optimum result with a traditional

optimization technique. Therefore, they used a nontraditional optimization technique

called as Genetic Algorithm. However, they also used the traditional optimization

methods to compare the traditional and nontraditional optimization techniques. They

concluded that genetic algorithm gives more accurate and more optimal results than

4

the traditional approach. They stated that since genetic algorithm is random search and

optimization technique, the chance of getting global optimum is possible. The weak

side of their study is that they used very simple formulas for contact stress and bending

stress evaluations. The contact stress formula proposed by them does not cover

Hertzian Contact theory. Bending stress formula proposed in their study does not

cover the geometry factor evaluated in AGMA standards. Therefore, geometry of the

gear root was not included in their study. Another weak side of their study is that they

didn’t clarify how the possibility of finding a local optimum is vanished. They also

didn’t evaluate the scuffing temperature. Saxena et al. [2] investigated the

optimization of internal spur gear design by using genetic Algorithm. They specified

the minimum center distance as the objective function. They specified some limits on

the contact ratio, gear ratio and tip interference. They also proposed a requirement to

eliminate the involute interference. They examined the face width as a constraint

function. They determined the required face width which meets bending and contact

stress requirements. The weak side of the study is that they used some charts to

evaluate the face width for the bending stress and for the contact stress. However, the

proposed charts do not include any information about the tooth geometry. Therefore,

the proposed formulas for face width evaluation can give the same results for the gears

which have different geometric shape with respect to each other. They stated that the

global optimum result was found by using genetic algorithm. However, Wu [3] stated

that genetic algorithm can give the local optimum. To overcome this issue, he

proposed an algorithm includes an improved genetic and simulated annealing

algorithm with the function of disturbance. Saxena et al. didn’t propose an improved

genetic algorithm and they stated that the global optimum was reached in their study.

They didn’t clarify how the possibility of finding a local optimum is vanished by using

genetic algorithm. Revar et al. [4] investigated the optimization of helical gears. They

considered the face width and helix angle as design variables. They changed the helix

angle and face width by keeping the other parameters constant. They evaluated the

maximum bending stress and the maximum contact stress by trial and error method.

They tried to find the optimum design that gives the maximum contact stress and

5

bending stress in certain limits. They made finite element analysis in ANSYS. They

concluded the effect of the helix angle and the face width on the contact stress and

bending stress. They stated that less thickness and helix angle give better and

maximum contact stress and bending stress. The weak side of the study is that they

didn’t use any optimization technique to obtain the optimum result. They tried to find

optimum solution by trial and error method. They considered a few parameters as

design parameters and they didn’t consider any geometrical constraint. However, they

used AGMA standards to evaluate the bending stress and contact stress. As compared

to [1] and [2], their results in terms of contact stress and bending stress were more

accurate than the results of the others since the gear tooth geometry was included.

Kapelevich and Shekhtman [5] examined the direct gear design. The direct gear design

is a gear design which is not restricted by a choice of gear tooth profiles based on

standard tool parameters. The method uses non-standard tooth shapes to cover the

custom applications. They investigated fillet optimization of standard rack-generated

gears by considering the bending stress. They tried to minimize the bending stress of

the standard rack-generated gears by applying the fillet optimization. They proposed

a method for balancing the bending stress of the pinion and the gear. They changed

the tooth thickness ratio in FEA program to satisfy the bending stress balance

requirement. They used FEA for bending stress evaluation because the Lewis equation

and its related coefficients do not provide a reliable solution to the wide variety of

non-standard gear tooth profiles. They concluded that optimization of the fillet profile

allowed reducing the maximum bending stress in the gear tooth root area by 10-30 %.

Padmanabhan et al. [6] investigated the optimization of spur gear pair by using

Genetic Algorithm. They defined the objective functions of the problem as follows:

maximization of power delivered by the gear pair, minimization of the overall weight,

maximization of the efficiency of the gear pair, minimization of the center distance.

They named the contact stress as the crushing stress. They took the crushing stress and

the bending stress as the constraint functions. Module, face width, number of teeth in

pinion and power were considered as design variables. They used the same formulas

for the bending stress and for the contact stress proposed in [1]. They verified the

6

results by implementing FEA. By FEA analysis the maximum stress was less than the

theoretical value. The weak side of the study is that bending stress and contact stress

formula used in their study didn’t include any information on the tooth geometry. They

didn’t consider the scuffing as a constraint function. Gopal et al. [7] investigated the

optimization of helical gear design by using Genetic Algorithm. The minimum center

distance was taken as objective function. They specified number of teeth on the pinion,

diametral pitch and helix angle as design variables. They took involute interference,

bending stress, contact stress, gear ratio and addendum to dedendum ratio as the design

variables. They used AGMA standards to evaluate the bending stress and contact

stress calculation. Their study didn’t neglect the geometry of the gear tooth. However,

they didn’t use the arbitrary addendum and dedendum radii. Therefore, their study is

limited to the gears which are restricted by the standard tool parameters. They didn’t

consider the scuffing evaluation. They didn’t suggest any proposal how the possibility

of converging a local optimum is vanished. Dr. Rajiv Suman et al. [8] investigated the

optimization of helical gears by using the Genetic Algorithm. Face width, center

distance and radius of the pinion were chosen as the parameters are to be optimized.

They evaluated the contact stress, bending stress and the involute interference as the

constraint functions. Pinion number of teeth, diametral pitch, helix angle, addendum

ratio and dedendum ratio were specified as the design parameters. Since they

considered the addendum and dedendum ratio as the design parameters, the proposed

method in their study does not include the direct gear design. Rai et al. [9] investigated

the optimization of spur gears by considering the center distance as the objective

function. Contact ratio, involute interference, bending stress, contact stress and

scuffing were considered as the constraint functions. They evaluated the optimization

problem with scuffing evaluation and without scuffing evaluation and they tried to

understand the effect of the scuffing into the optimized gear pair. They also used two

different optimization techniques in their study. They used the simulated annealing

and the real coded genetic algorithm for the same optimization problems. One of the

main interests of the study is comparison between these two techniques. Summary of

the some studies conducted in the literature is given in Table 1.1.

7

Table 1.1 Summary of some studies

Study Design Variables Method Constraints Geometry

Factor

[1]

Module

Mumber of teeth

of pinion

Genetic Algorithm Bending

Stress Contact

Stress

Not

included

[2]

Module

Number of teeth

of pinion

Genetic Algorithm Contact Ratio

Gear Ratio

Tip

Interference

Not

included

[6]

Module

Number of teeth

Power

Genetic Algorithm Crushing

stress

Bending stress

Not

included

[7]

Number of teeth,

Diametral pitch

Helix angle

Genetic Algorithm Involute

interference

Bending stress

Contact stress

Gear ratio

Addendum to

dedendum

ratio

Included

[8]

Module

Radius of the

pinion Diametral

pitch

Helix angle

Addendum ratio

Dedendum ratio

Genetic Algorithm Contact Stress

Bending Stress

Involute

interference

Included

[9]

Diametral pitch

Number of teeth

Simulated annealing

Real coded genetic

algorithm

Contact Stress

Involute

interference

Bending Stress

Scuffing

Temperature

Included

9

CHAPTER 2

2. CYLINDIRICAL EXTERNAL AND INTERNAL GEAR PAIR OPTIMIZATION

PARAMETERS AND OPTIMIZION METHODOLOGY

2.1. Cylindrical External Gear Pair Optimization Parameters

The optimization parameters for external and internal gear pairs are given in this

chapter. The optimization methodology is consisted of two constraint groups. The first

constraint group is named as geometrical constraints and the second constraint group

is named as material based constraints.

2.1.1. Geometrical Constraints

Geometrical constraints are investigated in this chapter. The geometrical constraints

are root clearance, top land thickness, contact ratio, involute clearance and tiff

clearance. The basic radii are given in Figure 2-1 and Figure 2-2.

Figure 2-1 Geometrical parameters of internal gear

10

Figure 2-2 Geometrical parameters of external gear

2.1.1.1. Root Clearance

Root clearance for the cylindrical external gear pairs is given in [11]. As seen from

Figure 2-3, the root clearance of the pinion and the gear is:

𝑐1 = 𝑎𝑤 − 𝑟𝑓1 − 𝑟𝑎2 (2.1)

𝑐2 = 𝑎𝑤 − 𝑟𝑓2 − 𝑟𝑎1 (2.2)

11

Figure 2-3 Root clearances 𝑐1 and 𝑐2 of an external gear pair [11]

Root clearance for the cylindrical internal gear pairs can be evaluated from Figure 2-4.

As seen from Figure 2-4, the root clearance of the pinion and the gear is:

𝑐1 = 𝑟𝑎2 − 𝑎𝑤 − 𝑟𝑓1 (2.3)

𝑐2 = 𝑟𝑓2−𝑎𝑤 − 𝑟𝑎1 (2.4)

Figure 2-4 Root clearances 𝑐1 and 𝑐2 of an internal gear pair

12

2.1.1.2. Top Land Thickness Evaluation of External Gear Pairs

The basic geometric relations and top land thickness evaluation is outlined in this

section. Basic geometric relations are given in Table 2.1.

Table 2.1 Basic geometric relations

Transverse circular pitch [12] 𝑝𝑡 =

𝑝𝑛

𝑐𝑜𝑠𝛽 (2.5)

Transverse tooth thickness in reference circle [12] 𝑠𝑡 =

𝑠𝑛

𝑐𝑜𝑠𝛽

(2.6)

Transverse module [12] 𝑚𝑡 =

𝑚𝑛

𝑐𝑜𝑠𝛽

(2.7)

Diameter of the reference cylinder [12] 𝑑 =

𝑧𝑚𝑛

𝑐𝑜𝑠𝛽

(2.8)

Normal base pitch [13] 𝑃𝑏𝑛 = 𝜋𝑚𝑛𝑐𝑜𝑠𝛼𝑛

(2.9)

Transverse base pitch [13] 𝑃𝑏𝑡 =

2𝜋𝑟𝑏1𝑡

𝑧1

(2.10)

Gear ratio [13] 𝑚𝐺 =

𝑧2

𝑧1

(2.11)

Base helix angle [13] 𝛽𝑏 = 𝑎𝑟𝑐𝑐𝑜𝑠 (

𝑃𝑏𝑛

𝑃𝑏𝑡)

(2.12)

Reference center distance [12] 𝑎𝑟𝑒𝑓 =

(𝑑1 + 𝑑2)

2

(2.13)

Profile shift coefficient [12]

𝑥 =𝑦

𝑚𝑛

(2.14)

Reference radius of the pinion [12] 𝑟1 =𝑧1𝑚𝑛

2𝑐𝑜𝑠𝛽= 𝑟1𝑢

(2.15)

Base radius of the pinion [12] 𝑟𝑏1 = 𝑟1𝑐𝑜𝑠𝛼𝑡

(2.16)

Base radius of the gear [12] 𝑟𝑏2 = 𝑟2𝑐𝑜𝑠𝛼𝑡

(2.17)

Transverse pressure angle [12] 𝛼𝑡 = 𝑎𝑟𝑐𝑡𝑎𝑛 (

𝑡𝑎𝑛𝛼𝑛

𝑐𝑜𝑠𝛽)

(2.18)

13

The normal circular tooth thickness of the zero backlash external gear at its reference

cylinder is: [12]

𝑆𝑛 =1

2𝜋𝑚𝑛 + 2𝑦𝑡𝑎𝑛𝛼𝑛

(2.19)

The angle between the normal plane and the transverse plane is equal to helix angle.

Therefore, tooth thickness at gear reference cylinder in transverse plane can be

expressed as follows:

𝑆𝑡 =𝑆𝑛

𝑐𝑜𝑠𝛽

(2.20)

If two external gears are to mesh with no backlash, their profile shift values must

satisfy: [12]

𝑦1 + 𝑦2 =𝑎𝑟𝑒𝑓(𝑖𝑛𝑣𝛼𝑤𝑡 − 𝑖𝑛𝑣𝛼𝑡)

𝑡𝑎𝑛𝛼𝑡

(2.21)

𝛼𝑤𝑡 = 𝑎𝑟𝑐𝑐𝑜𝑠 (𝑎𝑟𝑒𝑓𝑐𝑜𝑠𝛼𝑡

𝑎𝑤)

(2.22)

𝑖𝑛𝑣𝛼𝑡 = 𝑡𝑎𝑛𝛼𝑡 − 𝛼𝑡

(2.23)

𝑖𝑛𝑣𝛼𝑤𝑡 = 𝑡𝑎𝑛𝛼𝑤𝑡 − 𝛼𝑤𝑡

(2.24)

Sum of profile shift coefficients for zero backlash is:

𝑥1 + 𝑥2 =𝑎𝑟𝑒𝑓(𝑖𝑛𝑣𝛼𝑤𝑡 − 𝑖𝑛𝑣𝛼𝑡)

𝑚𝑛𝑡𝑎𝑛𝛼𝑡

(2.25)

Pinion operating radius is:

𝑟𝑤1 =𝑟𝑏1

𝑐𝑜𝑠𝛼𝑤𝑡

(2.26)

Gear operating radius is:

𝑟𝑤2 =𝑟𝑏2

𝑐𝑜𝑠𝛼𝑤𝑡

(2.27)

14

Figure 2-5 Tooth thickness of external gear pair pinion at reference and operating

circles

𝜙1 =𝑆𝑡1

𝑟1− 2(𝑖𝑛𝑣𝛼𝑤𝑡 − 𝑖𝑛𝑣𝛼𝑡)

(2.28)

𝑆𝑤𝑡1 = 𝑟𝑤1𝜙1 (2.29)

Backlash at operating circle in traverse section is given as:

𝐽𝑡𝑤𝑟 =𝐽𝑛𝑤𝑟

𝑐𝑜𝑠𝛽

(2.30)

Tooth thickness at operating circle with backlash in transverse section is given as:

𝑆𝑤𝑡1𝑏 = 𝑆𝑤𝑡1 −𝐽𝑡𝑤𝑟

2

(2.31)

Tooth thickness at reference circle with backlash in transverse section and in normal

section is given as:

𝜃1 = (𝑆𝑤𝑡1𝑏

𝑟𝑤1) + 2(𝑖𝑛𝑣𝛼𝑤𝑡 − 𝑖𝑛𝑣𝛼𝑡)

(2.32)

15

𝑆𝑡1𝑏 = 𝑟1𝜃1 = 𝑟1 ((𝑆𝑤𝑡1𝑏

𝑟𝑤1) + 2(𝑖𝑛𝑣𝛼𝑤𝑡 − 𝑖𝑛𝑣𝛼𝑡))

(2.33)

𝑆𝑛1𝑏 = 𝑆𝑡1𝑏𝑐𝑜𝑠𝛽 (2.34)

Figure 2-6 Top land thickness evaluation of external gear pair pinion

Top land thickness of the pinion in the transverse section is expressed as:

𝑠𝑛𝑎𝑡1 = 𝑟𝑎1(𝜃1 − 2(𝑖𝑛𝑣a1 − 𝑖𝑛𝑣𝛼𝑡))

(2.35)

Top land thickness of the pinion in the normal section is expressed as:

𝑠𝑛𝑎𝑛1=𝑠𝑛𝑎𝑡1𝑐𝑜𝑠 (𝑎𝑟𝑐𝑡𝑎𝑛 (𝑡𝑎𝑛𝛽𝑟𝑎1

𝑟1))

(2.36)

Similarly for the gear:

𝜙2 =𝑆𝑡2

𝑟2− 2(𝑖𝑛𝑣𝛼𝑤𝑡 − 𝑖𝑛𝑣𝛼𝑡)

(2.37)

𝑆𝑤𝑡2 = 𝑟𝑤2ψ2 (2.38)

16

Tooth thickness at operating circle with backlash in transverse section is given as:


2

(2.39)

Tooth thickness at reference circle with backlash in transverse section is given as:


𝑟𝑤2) + 2(𝑖𝑛𝑣𝛼𝑤𝑡 − 𝑖𝑛𝑣𝛼𝑡)

(2.40)

𝑆𝑡2𝑏 = 𝑟2𝜃2 = 𝑟2 ((𝑆𝑤𝑡2𝑏

𝑟𝑤2) + 2(𝑖𝑛𝑣𝛼𝑤𝑡 − 𝑖𝑛𝑣𝛼𝑡))

(2.41)

Top land thickness of the gear in the transverse section is expressed as:

𝑠𝑛𝑎𝑡2 = 𝑟𝑎2(𝜃2 − 2(𝑖𝑛𝑣a2 − 𝑖𝑛𝑣𝛼𝑡))

(2.42)

Top land thickness of the gear in the normal section is expressed as:

𝑠𝑛𝑎𝑛2 = 𝑠𝑛𝑎𝑡2𝑐𝑜𝑠 (𝑎𝑟𝑐𝑡𝑎𝑛 (𝑡𝑎𝑛𝛽𝑟𝑎2

𝑟2))

(2.43)

2.1.1.3. Top Land Thickness Evaluation of Internal Gear Pairs

The normal circular tooth thickness of the zero backlash internal gear at its reference

cylinder is: [12]

𝑆𝑛 =1

2𝜋𝑚𝑛 ± 2𝑦𝑡𝑎𝑛𝛼𝑛

(2.44)

If an external pinion and an internal gear are to mesh with no backlash, their profile

shift values must satisfy: [12]

𝑦2 − 𝑦1 =𝑎𝑟𝑒𝑓(𝑖𝑛𝑣𝛼𝑤𝑡 − 𝑖𝑛𝑣𝛼𝑡)

𝑡𝑎𝑛𝛼𝑡

(2.45)

Reference center distance for an internal gear pair is:

𝑎𝑟𝑒𝑓 =(𝑑2 − 𝑑1)

2= 𝑟2 − 𝑟1

(2.46)

17

Profile shift coefficients for zero backlash have the following relation:

𝑥2 − 𝑥1 =𝑎𝑟𝑒𝑓(𝑖𝑛𝑣𝛼𝑤𝑡 − 𝑖𝑛𝑣𝛼𝑡)

𝑚𝑛𝑡𝑎𝑛𝛼𝑡

(2.47)

From Figure 2-7,

𝜙2 =𝑆𝑡2

𝑟2− 2(𝑖𝑛𝑣𝛼𝑡 − 𝑖𝑛𝑣𝛼𝑤𝑡)

(2.48)

𝑆𝑤𝑡2 = 𝑟𝑤2𝜙2 (2.49)


2

(2.50)

Figure 2-7 Tooth thickness of internal gear at reference and operating circles

18

Tooth thickness at reference circle with backlash in transverse section is given as:


𝑟𝑤2) + 2(𝑖𝑛𝑣𝛼𝑡 − 𝑖𝑛𝑣𝛼𝑤𝑡)

(2.51)

𝑆𝑡2𝑏 = 𝑟2 ((𝑆𝑤𝑡2𝑏

𝑟𝑤2) + 2(𝑖𝑛𝑣𝛼𝑡 − 𝑖𝑛𝑣𝛼𝑤𝑡))

(2.52)

𝑆𝑛2𝑏 = 𝑆𝑡2𝑏𝑐𝑜𝑠𝛽

(2.53)

Top land thickness of the gear in the transverse section is expressed as:

𝑠𝑛𝑎𝑡2 = 𝑟𝑎2(𝜃2 − 2(𝑖𝑛𝑣𝛼𝑡 − 𝑖𝑛𝑣a2)) (2.54)

Top land thickness of the gear in the normal section is expressed as:

𝑠𝑛𝑎𝑛2 = 𝑠𝑡𝑜𝑝𝑙𝑎𝑛𝑑𝑡2𝑐𝑜𝑠 (𝑎𝑟𝑐𝑡𝑎𝑛 (𝑡𝑎𝑛𝛽𝑟𝑎2

𝑟2))

(2.55)

Figure 2-8 Top land thickness evaluation of internal gear pair

19

2.1.1.4. Contact Ratio Evaluation of External Gear Pairs

Tip form radius of the pinion of external gear pair is given as:

𝑟𝑡𝑓1 = 𝑟𝑎1 − 𝑡𝑐 (2.56)

Tip form radius of the gear of external gear pair is given as:

𝑟𝑡𝑓2 = 𝑟𝑎2 − 𝑡𝑐 (2.57)

Figure 2-9 Transverse plane view of the line of action of an external gear pair

20

𝐶6 = (𝑟𝑏1 + 𝑟𝑏2)𝑡𝑎𝑛𝛼𝑤𝑡 = 𝑎𝑤𝑠𝑖𝑛𝛼𝑤𝑡 (2.58)

𝐶1 = 𝐶6 − √𝑟𝑡𝑓22 − 𝑟𝑏2

2

(2.59)

𝐶5 = √𝑟𝑡𝑓12 − 𝑟𝑏1

2

(2.60)

𝐶2 = 𝐶5 − 𝑃𝑏𝑡 (2.61)

𝐶3 = 𝑟𝑏1𝑡𝑎𝑛𝛼𝑤𝑡 (2.62)

𝐶4 = 𝐶1 + 𝑃𝑏𝑡 (2.63)

𝑟𝑆𝐴𝑃1= √𝑟𝑏1

2 + 𝐶12

(2.64)

𝑟𝑆𝐴𝑃2= √𝑟𝑏2

2 + (𝐶6 − 𝐶5)2

(2.65)

Active length of line of action is expressed as:

𝑍 = 𝐶5 − 𝐶1 (2.66)

Transverse contact ratio is given as: [13]

𝑚𝑝 =𝑍

𝑃𝑏𝑡

(2.67)

2.1.1.5. Contact Ratio Evaluation of Internal Gear Pairs

Contact ratio evaluation of the internal gear pair is conducted as follows:

Tip form radius of the pinion of internal gear pair is given as:

𝑟𝑡𝑓1 = 𝑟𝑎1 − 𝑡𝑐 (2.68)

Tip form radius of the gear of internal gear pair is given as:

𝑟𝑡𝑓2 = 𝑟𝑎2 + 𝑡𝑐 (2.69)

21

Figure 2-10 Transverse plane view of the line of action of an internal gear pair

𝐶6 = (𝑟𝑏2 − 𝑟𝑏1)𝑡𝑎𝑛𝛼𝑤𝑡 = 𝑎𝑤𝑠𝑖𝑛𝛼𝑤𝑡 (2.70)

𝐶1 = √𝑟𝑡𝑓22 − 𝑟𝑏2

2 − 𝐶6

(2.71)

𝐶5 = √𝑟𝑡𝑓12 − 𝑟𝑏1

2

(2.72)

22

𝐶2 = 𝐶5 − 𝑃𝑏𝑡 (2.73)

𝐶3 = 𝑟𝑏1𝑡𝑎𝑛𝛼𝑤𝑡 (2.74)

𝐶4 = 𝐶1 + 𝑃𝑏𝑡 (2.75)

Active length of line of action is expressed as:

𝑍 = 𝐶5 − 𝐶1 (2.76)

Transverse contact ratio is given as: [13]

𝑚𝑝 =𝑍

𝑃𝑏𝑡

(2.77)

2.1.1.6. Gear Root Evaluation of External Gear Pairs

Gear root evaluation of the external gear pairs is conducted in this section. Helical

gears have rounded gear root in normal section. Therefore, gear root evaluation is

evaluated in transverse section and then it is transmitted into the normal section. As

seen from Figure 2-11,

𝜁1 = [(𝑝𝑡−𝑆𝑡1𝑏

2 )

𝑟1− (𝑖𝑛𝑣𝛼𝑡 − 𝑖𝑛𝑣𝛼𝑅𝑓1𝑡

)]

(2.78)

𝑅𝑓1𝑡𝑥 = 𝑅𝑓1𝑡𝑠𝑖𝑛𝜁1

(2.79)

𝑅𝑓1𝑡𝑦 = 𝑅𝑓1𝑡𝑐𝑜𝑠𝜁1

(2.80)

Conversion of the 𝑅𝑓1𝑡 into the normal plane is given as

𝑅𝑓1𝑛𝑥 = 𝑅𝑓1𝑡𝑠𝑖𝑛𝜁1𝑐𝑜𝑠𝛽𝑅𝑓1𝑡

(2.81)

𝑅𝑓1𝑛𝑦 = 𝑅𝑓1𝑡𝑐𝑜𝑠𝜉1

(2.82)

𝑅𝑓1𝑛 = √𝑅𝑓1𝑛𝑥2 + 𝑅𝑓1𝑛𝑦

2

(2.83)

As seen from Figure 2-11,

23

𝑟𝑏1𝑡𝑥 = 𝑟𝑏1𝑠𝑖𝑛 (𝜁1 + 𝛼𝑅𝑓1𝑡)

(2.84)

𝑟𝑏1𝑡𝑦 = 𝑟𝑏1𝑐𝑜𝑠 (𝜁1 + 𝛼𝑅𝑓1𝑡)

(2.85)

Conversion into the normal section

𝑟𝑏1𝑛𝑥 = 𝑟𝑏1𝑠𝑖𝑛 (𝜁1 + 𝛼𝑅𝑓1𝑡) 𝑐𝑜𝑠𝛽𝑟𝑓1𝑡

(2.86)

𝑟𝑏1𝑛𝑦 = 𝑟𝑏1𝑐𝑜𝑠 (𝜁1 + 𝛼𝑅𝑓1𝑡)

(2.87)

𝑟𝑏1𝑛 = √𝑟𝑏1𝑛𝑥2 + 𝑟𝑏1𝑛𝑦

2

(2.88)

Figure 2-11 Transverse plane view of external gear pair pinion tooth

As seen from Figure 2-12, set of equation of the gear root is given as:

𝜃𝑥1 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝑎1

𝑏1)

(2.89)

𝑅𝑓1𝑛𝑦 + 𝐴1𝑐𝑜𝑠(𝜃1 − 𝜃𝑥1) − 𝑏1 = 0 (2.90)

24

𝑎1 + 𝐴1𝑠𝑖𝑛(𝜃1 − 𝜃𝑥1) − 𝑅𝑓1𝑛𝑥 = 0 (2.91)

𝑎12 + 𝑏1

2 − (𝑟𝑓1 + 𝐴1)2

= 0 (2.92)

√(𝑟𝑓1 + 𝐴1)2− 𝑟𝑏1𝑛

2 − 𝐴1 − √𝑅𝑓1𝑛2 − 𝑟𝑏1𝑛

2 = 0 (2.93)

𝐴1, 𝜃1, 𝑅𝑓1𝑛, 𝑏1 are unknown parameters which are to be determined by solving the

nonlinear equations given above. Initial values are assigned for these unknown

parameters and the set of equation is solved by using an iterative method. In this study,

the gear root equations are solved by using fsolve method in MATLAB.

Figure 2-12 Normal plane view of external gear pair pinion tooth

The gear root evaluation for the external gear is conducted by using the same

methodology given previously for external pinion.

25

2.1.1.7. Gear Root Evaluation of Internal Gear Pairs

Gear root evaluation of the internal gear pairs is conducted in this section. Gear root

evaluation of the external gear pairs are given in the previous section. The same

methodology given in Chapter 2.1.1.6 is used for the gear root evaluation of the pinion

of internal gear pairs. Helical gears have rounded gear root in normal section.

Therefore, gear root evaluation is evaluated in transverse section and then it is

transmitted into the normal section. As seen from Figure 2-13,

𝜁2 = [(𝑝𝑡−𝑆𝑡2𝑏

2)

𝑟2− (𝑖𝑛𝑣𝛼𝑅𝑓2𝑡

− 𝑖𝑛𝑣𝛼𝑡)]

(2.94)

𝑅𝑓2𝑡𝑥 = 𝑅𝑓2𝑡𝑠𝑖𝑛𝜁2

(2.95)

𝑅𝑓2𝑡𝑦 = 𝑅𝑓2𝑡𝑐𝑜𝑠𝜁2

(2.96)

Conversion of the 𝑅𝑓2𝑡 into the normal plane is given as

𝑅𝑓2𝑛𝑥 = 𝑅𝑓2𝑡𝑠𝑖𝑛𝜁2𝑐𝑜𝑠𝛽𝑅𝑓2𝑡

(2.97)

𝑅𝑓2𝑛𝑦 = 𝑅𝑓2𝑡𝑐𝑜𝑠𝜉2

(2.98)

𝑅𝑓2𝑛 = √𝑅𝑓2𝑛𝑥2 + 𝑅𝑓2𝑛𝑦

2

(2.99)


𝑟𝑏2𝑡𝑥 = 𝑟𝑏2𝑠𝑖𝑛 (𝛼𝑅𝑓2𝑡− 𝜁2)

(2.100)

𝑟𝑏2𝑡𝑦 = 𝑟𝑏2𝑐𝑜𝑠 (𝛼𝑅𝑓2𝑡− 𝜁2)

(2.101)

Conversion into the normal section

𝑟𝑏2𝑛𝑥 = 𝑟𝑏2𝑠𝑖𝑛 (𝛼𝑅𝑓2𝑡− 𝜁2) 𝑐𝑜𝑠𝛽𝑅𝑓2𝑡

(2.102)

𝑟𝑏2𝑛𝑥 = 𝑟𝑏2𝑐𝑜𝑠 (𝛼𝑅𝑓2𝑡− 𝜁2)

(2.103)

𝑟𝑏2𝑛 = √𝑟𝑏2𝑛𝑥2 + 𝑟𝑏2𝑛𝑦

2

(2.104)

26

Figure 2-13 Transverse plane view of internal gear pair gear tooth

As seen from Figure 2-14, the set of equation of the gear root is given as:


𝑏2)

(2.105)

𝑅𝑓2𝑛𝑦 − 𝐴2𝑐𝑜𝑠(𝜃2 + 𝜃𝑥2) − 𝑏2 = 0

(2.106)

𝑎2 + 𝐴2𝑠𝑖𝑛(𝜃2 + 𝜃𝑥2) − 𝑅𝑓2𝑛𝑥 = 0

(2.107)

𝑎22 + 𝑏2

2 − (𝑟𝑑2 − 𝐴2)2 = 0

(2.108)

√(𝑟𝑏2𝑛𝑡𝑎𝑛 (𝜋

2− (𝜃2 + 𝜃𝑥2)) +

𝑎2

𝑠𝑖𝑛(𝜃2 + 𝜃𝑥2)+ 𝐴2)

2

+ 𝑟𝑏2𝑛2

− √𝑅𝑓2𝑛𝑥2 + 𝑅𝑓2𝑛𝑦

2 = 0

(2.109)

27

𝐴2, 𝜃2, 𝑅𝑓2𝑛, 𝑏2 are unknown parameters which are to be determined by solving the

nonlinear equations given above. Initial values are assigned for these unknown

parameters and the set of equation is solved by using an iterative method. In this study,

the gear root equations are solved by using fsolve method in MATLAB.

Figure 2-14 Normal plane view of internal gear pair gear tooth

2.1.1.8. Involute Clearance

The gear tooth should have a clearance between the starting radius of the involute and

the base radius to eliminate the mounting and manufacturing errors. Root form radius

which is lower than the base radius causes the undercutting of the gear root. The

clearance is mathematically expressed for the pinion of external and the internal gear

pairs as:

28

𝐼𝑐1 = 𝑅𝑓1 − 𝑟𝑏1 (2.110)

For the gear of external gear as:

𝐼𝑐2 = 𝑅𝑓2 − 𝑟𝑏2 (2.111)

For internal gears the clearance between the starting radius of the involute and the

base radius is expressed as:

𝐼𝑐2 = 𝑟𝑎2 − 𝑟𝑏2 (2.112)

2.1.1.9. Tiff Clearance

Clearance between the start of active profile radius and the root form radius is defined

as the tiff clearance. Tiff clearance is a required clearance to satisfy the operating

condition of the gear pair. The start of active profile radius which is lower than the

root form radius causes the crashing of the pinion and gear teeth and the gear pair

could not operate. The clearance for the pinion of the external and the internal gear

pairs is mathematically expressed as:

𝑇𝑖𝑓𝑓1 = 𝑟𝑆𝐴𝑃1− 𝑅𝑓1

(2.113)

Tiff clearance for the gear of external gear pairs is:

𝑇𝑖𝑓𝑓2 = 𝑟𝑆𝐴𝑃2− 𝑅𝑓2

(2.114)

Tiff clearance for the gear of internal gear pairs is:

𝑇𝑖𝑓𝑓2 = 𝑅𝑓2 − 𝑟𝑆𝐴𝑃2

(2.115)

2.1.2. Material Based Constraints

Material based constraints are investigated in this chapter. The material based

constraints are consisted of three groups. These are conducted as follow:

Contact stress number evaluation, bending stress number evaluation and scuffing risk

evaluation.

29

2.1.2.1. Contact Stress Number Evaluation of External Gear Pairs

The contact stress number formula for gear teeth is: [14]

𝑠𝑐 = 𝐶𝑝√𝑊𝑡𝐾𝑜𝐾𝑣𝐾𝑠

𝐾𝑚

𝑑𝑤1𝐹

𝐶𝑓

𝐼

(2.116)

Contact strength geometry factor is given as: [13]

𝐼 =𝑐𝑜𝑠𝛼𝑡𝐶𝜓

2

(1𝜌1

±1𝜌2

)𝑑𝑤1𝑚𝑁

(2.117)

Helical overlap factor, 𝐶𝜓 is given as follows: [13]

For LACR helical gears (𝑚𝐹 ≤ 1.0)

𝐶𝜓 = [1 − 𝑚𝐹 (1 −𝜌𝑚1𝜌𝑚2𝑍

𝜌1𝜌2𝑃𝑏𝑛)]

0.5

(2.118)

Radius of curvature of the pinion profile at the mean radius of the pinion, 𝜌𝑚1

𝜌𝑚1 = (𝑅𝑚1 − 𝑟𝑏1)0.5

(2.119)

Radius of curvature of the gear profile at the mean radius of the gear, 𝜌𝑚2

𝜌𝑚2 = 𝐶6 ∓ 𝜌𝑚1 (2.120)

For spurs and conventional helical gears (𝑚𝐹 > 1.0)

𝐶𝜓 = 1.0 (2.121)

Load sharing ratio, 𝑚𝑛 is given as follows:

For helical gears:

𝑚𝑁 =𝐹

𝐿𝑚𝑖𝑛

(2.122)

For spur gears with 𝑚𝑃 < 2.0, gives 𝐿𝑚𝑖𝑛 = 𝐹, therefore:

30

𝑚𝑁 = 1.0 (2.123)

For LACR helicals, (𝑚𝐹 ≤ 1.0), load sharing is accommodated by 𝐶𝜓, therefore load

sharing factor is taken as unity.

Face width ratio is given in Table 2.2 according to gear mounting relative to bearings.

Table 2.2 Maximum Face width Ratio [15]

In the current study face width ratio is taken as ‘’0.7’’. Therefore, the face width ratio

considered in this study is applicable for spur and helical gears which are induction

hardened or case carburized with lead correction or without lead correction. The face

width is given as:

𝐹 = 0.7 ∙ 𝑚𝑡 ∙ 𝑁1 (2.124)

Axial contact ratio is given as:

𝑚𝐹 =𝐹

𝑃𝑥

(2.125)

31

where axial pitch, 𝑃𝑥 , is

𝑃𝑥 =𝜋

𝑠𝑖𝑛𝛽

(2.126)

Minimum length of the lines of contact is given as follows:

For spur gears with 𝑚𝑃 < 2.0 the minimum length of contact lines, 𝐿𝑚𝑖𝑛 , is

𝐿𝑚𝑖𝑛 = 𝐹 (2.127)

For helical gears, two cases must be considered:

Case I, for 𝑛𝑎 ≤ 1 − 𝑛𝑟

𝐿𝑚𝑖𝑛 =𝑚𝑃𝐹 − 𝑛𝑎𝑛𝑟𝑃𝑥

𝑐𝑜𝑠𝛽𝑏

(2.128)

Case II, for 𝑛𝑎 > 1 − 𝑛𝑟

𝐿𝑚𝑖𝑛 =𝑚𝑃𝐹 − (1 − 𝑛𝑎)(1 − 𝑛𝑟)𝑃𝑥


(2.129)

where 𝑛𝑟 is the fractional part of 𝑚𝑃 and 𝑛𝑎 is the fractional part of 𝑚𝐹.

Normal base pitch is given as:

𝑃𝑏𝑛 = 𝜋𝑚𝑛𝑐𝑜𝑠𝛼𝑛 (2.130)

Transverse base pitch

𝑃𝑏𝑡 =2𝜋𝑟𝑏1𝑡

𝑧1

(2.131)

Base helix angle

𝛽𝑏 = 𝑎𝑟𝑐𝑐𝑜𝑠 (𝑃𝑏𝑛

𝑃𝑏𝑡)

(2.132)

Radii of curvature of profiles at stress calculation point are given as follows:

32

For conventional helical gears (𝑚𝐹 > 1.0) the radii of curvature are calculated at the

mean radius or middle of the working profile of the pinion where:

Mean radius of pinion, 𝑅𝑚1

𝑅𝑚1 =1

2[𝑟𝑎1 ± (𝑎𝑤 − 𝑟𝑎2)]

(2.133)

Radius of curvature of the pinion profile at the point of contact stress calculation, 𝜌1

𝜌1 = √𝑅𝑚12 − 𝑟𝑏1

2

(2.134)

Radius of curvature of the gear profile at the point of contact stress calculation, 𝜌2

𝜌2 = 𝐶6 ∓ 𝜌1 (2.135)

For spurs and LACR helical gears (𝑚𝐹 ≤ 1.0) the radii of curvature are calculated at

the LPSTC

𝜌1 = 𝐶2 (2.136)

𝜌2 = 𝐶6 ∓ 𝜌1 (2.137)

2.1.2.2. Bending Strength Geometry Factor and Bending Stress Number

Evaluation

The fundamental formula for bending stress number is given in [14]

𝑠𝑡 = 𝑊𝑡𝐾𝑜𝐾𝑣𝐾𝑠

𝑃𝑑

𝐹

𝐾𝑚𝐾𝐵

𝐽

(2.138)

Bending stress geometry factor is evaluated as follows:

Bending stress geometry factor is given as: [13]

𝐽 =𝑌𝐶𝜓

𝐾𝑓𝑚𝑁

(2.139)

33

Tooth form factor is given in [13].

𝑌 =

𝐾𝜓

𝑐𝑜𝑠𝛼𝑛𝐿

𝑐𝑜𝑠𝛼𝑤[

6ℎ𝑓

𝑆𝐹2𝐶ℎ

−𝑡𝑎𝑛𝛼𝑛𝐿

𝑆𝐹]𝑚𝑛

(2.140)

Helical factor is given as follows: [13]

For Spur and LACR Helical Gears (𝑚𝐹 ≤ 1.0), a unity value is used,

𝐶ℎ = 1.0 (2.141)

For Conventional Helical Gears, when 𝑚𝐹 > 1.0

𝐶ℎ =1

1 − [𝑤

100 (1 −𝑤

100)]0.5

(2.142)

𝑤 = 𝑎𝑟𝑐𝑡𝑎𝑛(𝑡𝑎𝑛𝛽𝑠𝑖𝑛𝛼𝑛) (2.143)

The helix angle factor given in [13] depends on the type of gear. For Spur and LACR

Helical Gears (𝑚𝐹 ≤ 1.0), a unity value is used. For Conventional Helical Gears,

when 𝑚𝐹 > 1.0

𝐾𝜓 = 𝑐𝑜𝑠𝛽𝑤𝑐𝑜𝑠𝛽 (2.144)

Stress correction factor for the pinion is given as:

𝐾𝑓1 = 𝐻 + (𝑆𝐹1

𝐴1𝑣)𝐿

(𝑆𝐹1

ℎ𝑓1)

𝑀

(2.145)

Similarly for the gear

𝐾𝑓2 = 𝐻 + (𝑆𝐹2

𝐴2𝑣)

𝐿

(𝑆𝐹2

ℎ𝑓2)

𝑀

(2.146)

where 𝐻 = 0.331 − 0.436𝛼𝑛 𝐿 = 0.324 − 0.492𝛼𝑛 𝑀 = 0.261 + 0.545𝛼𝑛

34

Virtual Spur Gear Evaluation

Helical gears are considered as virtual spur gears with the following virtual geometry

while for the spur gear the actual geometry is used. The following geometrical

relations given in Table 2.3 are used for virtual gear root evaluation.

Table 2.3 Basic geometric relations of virtual gear

Virtual tooth number of the pinion

[13] 𝑧1𝑣 =

𝑧1

𝑐𝑜𝑠3𝛽

(2.147)

Virtual tooth number of the gear [13] 𝑧2𝑣 =

𝑧2

𝑐𝑜𝑠3𝛽 (2.148)

Virtual reference radius of the pinion

[13] 𝑟1𝑣 = 𝑚𝑛

𝑧1𝑣

2 (2.149)

Virtual reference radius of the gear

[13] 𝑟2𝑣 = 𝑚𝑛

𝑧2𝑣

2 (2.150)

Virtual base radius of the pinion [13] 𝑟𝑏1𝑣 = 𝑟1𝑣𝑐𝑜𝑠𝛼𝑛 (2.151)

Virtual base radius of the gear [13] 𝑟𝑏2𝑣 = 𝑟2𝑣𝑐𝑜𝑠𝛼𝑛 (2.152)

Virtual outside radius of the pinion

[13] 𝑟𝑎1𝑣 = 𝑟1𝑣 + 𝑟𝑎1 − 𝑟1 (2.153)

Virtual outside radius of the gear [13] 𝑟𝑎2𝑣 = 𝑟2𝑣 + 𝑟𝑎2 − 𝑟2 (2.154)

Virtual root radius of the pinion [13] 𝑟𝑓1𝑣 = 𝑟1𝑣 − 𝑟1 + 𝑟𝑓1 (2.155)

Virtual root radius of the gear [13] 𝑟𝑓2𝑣 = 𝑟2𝑣 − 𝑟2 + 𝑟𝑓2 (2.156)

Tip form radii of the pinion and the gear can be expressed in a similar way used in

evaluating the outside radii. Virtual tip form radius of the pinion:

𝑟𝑡𝑓1𝑣 = 𝑟1𝑣 + 𝑟𝑡𝑓1 − 𝑟1

(2.157)

Virtual tip form radius of the gear:

𝑟𝑡𝑓2𝑣 = 𝑟2𝑣 + 𝑟𝑡𝑓2 − 𝑟2

(2.158)

Sixth distance along line of action of virtual spur gear is:

𝐶6𝑣 = (𝑟𝑏2𝑣 ± 𝑟𝑏1𝑣)𝑡𝑎𝑛𝛼𝑤𝑛

(2.159)

35

First distance along line of action of virtual spur gear is:

𝐶1𝑣 = ± [𝐶6𝑣 − (𝑟𝑡𝑓2𝑣2 − 𝑟𝑏2𝑣

2)0.5

]

(2.160)

Fourth distance along line of action of virtual spur gear is:

𝐶4𝑣 = 𝐶1𝑣 + 𝑃𝑏𝑛

(2.161)

Load Angle and Load Radius for External Gear Pairs

For helical gears and spur gears that are analyzed where the load is applied at the tip

of the tooth, the pressure angle at load application point for the pinion is given in [13].

𝑡𝑎𝑛𝛼𝑛𝑊1 = [(𝑟𝑡𝑓1𝑣

𝑟𝑏1𝑣)2

− 1]

0.5

(2.162)

For spur gears, where the highest bending stress occurs when the load is at the highest

point of single tooth contact (HPSTC), the pressure angle for the pinion is given in

[13].

𝑡𝑎𝑛𝛼𝑛𝑊1 =𝐶4

𝑟𝑏1𝑣

(2.163)

Equation (2.163) may also be used for LACR helical gears (𝑚𝐹 ≤ 1.0), but distance

𝐶4 must be based on the virtual spur gear.

36

Figure 2-15 Pressure angle where external gear tooth comes to point

As seen from Figure 2-15.

𝑖𝑛𝑣𝛼𝑛𝑝1 = 𝑖𝑛𝑣𝛼𝑛 +𝑠𝑛1

2𝑟1𝑣

(2.164)

𝑖𝑛𝑣𝛼𝑛 = 𝑡𝑎𝑛𝛼𝑛 − 𝛼𝑛 (2.165)

By subs.

𝑖𝑛𝑣𝛼𝑛𝑝1 = 𝑡𝑎𝑛𝛼𝑛 − 𝛼𝑛 +𝑠𝑛1

2𝑟1𝑣

(2.166)


𝛼𝑛𝐿1 = 𝑡𝑎𝑛𝛼𝑛𝑤1 − 𝑖𝑛𝑣𝛼𝑛𝑝1 (2.167)

37

𝛼𝑛𝐿1 = 𝑡𝑎𝑛𝛼𝑛𝑤1 − 𝑡𝑎𝑛𝛼𝑛 + 𝛼𝑛 −𝑠𝑛1

2𝑟1𝑣

(2.168)

𝑟𝑛𝐿1 =𝑟𝑏1𝑣

𝑐𝑜𝑠𝛼𝑛𝐿1

(2.169)

Figure 2-16 Load angle and load radius of an external gear

Virtual Gear Root Evaluation for External Gear Pairs

Virtual gear root evaluation is outlined in this chapter. As seen from Figure 2-17,

𝑅𝑓1𝑣 = √(√(𝑟𝑓1𝑣 + 𝐴1𝑣)2− 𝑟𝑏1𝑣

2 − 𝐴1𝑣)

2

+ 𝑟𝑏1𝑣2

(2.170)

𝛾1𝑣 = 𝑎𝑟𝑐𝑐𝑜𝑠 (𝑟𝑏1𝑣

𝑟𝑓1𝑣 + 𝐴1𝑣)

(2.171)

38

𝛼𝑅𝑓1𝑣= 𝑎𝑟𝑐𝑐𝑜𝑠 (

𝑟𝑏1𝑣

𝑅𝑓1𝑣)

(2.172)

𝑖𝑛𝑣𝛼𝑅𝑓1𝑣= 𝑡𝑎𝑛𝛼𝑅𝑓1𝑣

− 𝛼𝑅𝑓1𝑣

(2.173)

𝜃1𝑣 = 𝑎𝑟𝑐𝑠𝑖𝑛 (𝑟𝑏1𝑣

𝑟𝑓1𝑣 + 𝐴1𝑣)

(2.174)

𝑏1𝑣 = (𝑟𝑓1𝑣 + 𝐴1𝑣)𝑐𝑜𝑠 (𝜃𝑝1𝑣 − (𝑖𝑛𝑣𝛼𝑛 − 𝑖𝑛𝑣𝛼𝑅𝑓1𝑣) − (𝛾1𝑣 − 𝛼𝑅𝑓1𝑣

))

(2.175)

𝜃𝑥1𝑣 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝑎1𝑣

𝑏1𝑣)

(2.176)

𝜃𝑝1𝑣 =(𝑝𝑛 − 𝑠𝑛1𝑏)

2𝑟1𝑣

(2.177)

Figure 2-17 Normal plane view of virtual spur gear

The eqution of the virtual gear root of the pinion is expressed as:

39

𝑎1𝑣 + 𝐴1𝑣𝑠𝑖𝑛(𝜃1𝑣 − 𝜃𝑥1𝑣) − 𝑅𝑓1𝑣𝑠𝑖𝑛 (𝜃𝑝1𝑣 − (𝑖𝑛𝑣𝛼𝑛 − 𝑖𝑛𝑣𝛼𝑅𝑓1𝑣

))

= 0

(2.178)

This nonlinear equation is solved by implementing an iterative method. The equation

is solved by using fsolve in MATLAB in this study. The virtual gear root evaluation

of the gear of external gear pair is conducted by using the same methodology which

is outlined in the current section.

Critical Section Determination for External Gear Pairs

The related equations for evaluation of the critical section are outlined in this section.


𝛼𝑐𝑟𝑣1 = arctan (𝑎1𝑣

𝑏1𝑣) + 𝑎𝑟𝑐𝑐𝑜𝑠

(

𝑎1𝑣

2 + 𝑏1𝑣2 + 𝑟𝑐𝑟𝑣1

2 − 𝐴1𝑣2

2𝑟𝑐𝑟𝑣1√𝑎1𝑣2 + 𝑏1𝑣

2

)

(2.179)

𝐿ℎ𝑣1 =𝑟𝑏1𝑣

cos (𝛼𝑛𝐿1)

(2.180)

ℎ𝑓1 = 𝐿ℎ𝑣1 − 𝑟𝑐𝑟𝑣1𝑐𝑜𝑠 (𝜋

𝑧1𝑣− 𝛼𝑐𝑟𝑣1)

(2.181)

𝑆𝐹1 = 2𝑟𝑐𝑟𝑣1𝑠𝑖𝑛 (𝜋


(2.182)

Similarly for virtual gear root



(

𝑎2𝑣

2 + 𝑏2𝑣2 + 𝑟𝑐𝑟𝑣2

2 − 𝐴2𝑣2


2

)

(2.183)

ℎ𝑓2 = 𝐿ℎ𝑣2 − 𝑟𝑐𝑟𝑣2𝑐𝑜𝑠 (𝜋


(2.184)



(2.185)

40

Figure 2-18 Critical section evaluation of virtual spur gear

Evaluation of the Critical Radius for External Gear Pairs

Critical radius is determined by using the following methodology outlined in this

section.

Table 2.4 Critical radius algorithm for the pinion

𝒇𝒐𝒓 𝒊 = 𝟏: 𝟓𝟎

𝑟𝑐𝑟𝑣1(1) = 𝑟𝑓1𝑣

𝑟𝑐𝑟𝑣1(𝑖 + 1) = 𝑟𝑐𝑟𝑣1(𝑖) +(𝑅𝑓1𝑣 − 𝑟𝑓1𝑣)

𝑖𝑡𝑟

𝒆𝒏𝒅

41

The same algorithm given in Table 2.4 is used to evaluate the critical radius of the

gear. The consisted radii are substituted into the formulas of 𝛼𝑐𝑟𝑣1, ℎ𝑓1 and 𝑆𝐹1. Then,

the related 𝑌1, 𝐾𝑓1 values are obtained for each different 𝛼𝑐𝑟𝑣1, ℎ𝑓1 and 𝑆𝐹1. For each

different 𝑌1, 𝐾𝑓1 values, the different geometry factor values of the pinion ( 𝐽1 ) are

evaluated. Mathematically it is expressed as:

Table 2.5 Algorithm of bending strength geometry factor for the pinion

𝒇𝒐𝒓 𝒊 = 𝟏: 𝟓𝟎

𝛼𝑐𝑟𝑣1(𝑖) = arctan (𝑎1𝑣


(

𝑎1𝑣

2 + 𝑏1𝑣2 + (𝑟𝑐𝑟𝑣1(𝑖))

2− 𝐴1𝑣

2

2𝑟𝑐𝑟𝑣1(𝑖)√𝑎1𝑣2 + 𝑏1𝑣

2

)

ℎ𝑓1(𝑖) = 𝐿ℎ𝑣1 − 𝑟𝑐𝑟𝑣1(𝑖)𝑐𝑜𝑠 (𝜋

𝑧1𝑣− 𝛼𝑐𝑟𝑣1(𝑖))

𝑌1(𝑖) =𝐾𝜓


𝑐𝑜𝑠𝛼𝑤1[

6ℎ𝑓1(𝑖)

(𝑆𝐹1(𝑖))2𝐶ℎ

−𝑡𝑎𝑛𝛼𝑛𝐿1

𝑆𝐹1(𝑖)]

𝑆𝐹1(𝑖) = 2𝑟𝑐𝑟𝑣1(𝑖)𝑠𝑖𝑛 (𝜋





6ℎ𝑓1(𝑖)



𝑆𝐹1(𝑖)]

𝐾𝑓1(𝑖) = 𝐻 + (𝑆𝐹1(𝑖)

𝐴1𝑣)

𝐿

(𝑆𝐹1(𝑖)

ℎ𝑓1(𝑖))

𝑀

𝐽1(𝑖) =𝑌1(𝑖)𝐶𝜓

𝐾𝑓1(𝑖)𝑚𝑁

𝒆𝒏𝒅

42

The same algorithm given in Table 2.5 is used for the bending stress calculation of the

gear.The minimum ones of the 𝐽1 and 𝐽2 are taken as the geometry factor and

implemented into the bending stress equation.

Load Angle and Load Radius for Internal Gear Pairs

The pressure angle at load application point for the pinion is given by:


𝑟𝑏1𝑣)2

− 1]

0.5

(2.186)

The pressure angle at load application for the gear is given by:


𝑟𝑏2𝑣)2

− 1]

0.5

(2.187)

For spur gears, where the highest bending stress occurs when the load is at the highest

point of single tooth contact (HPSTC), the pressure angle for the pinion is given by:


𝑟𝑏1𝑣

(2.188)

Pressure angle for the gear is given by:


𝑟𝑏2𝑣

(2.189)

Equation (2.188) and equation (2.189) May also be used for LACR helical gears

(𝑚𝐹 ≤ 1.0), but distance 𝐶4 must be based on the virtual spur gear.

The pressure angle for the pinion is:

𝑡𝑎𝑛𝛼𝑛𝑊1 =𝐶4𝑣

𝑟𝑏1𝑣

(2.190)

43

The pressure angle for the gear is:

𝑡𝑎𝑛𝛼𝑛𝑊2 =𝐶4𝑣

𝑟𝑏2𝑣

(2.191)

Figure 2-19 Pressure angle where internal gear tooth comes to point

44

Figure 2-20 Load angle and load radius of an internal gear


𝑖𝑛𝑣𝛼𝑛𝑝2 =𝑠𝑛2

2𝑟2𝑣− 𝑖𝑛𝑣𝛼𝑛

(2.192)

𝑖𝑛𝑣𝛼𝑛 = 𝑡𝑎𝑛𝛼𝑛 − 𝛼𝑛 (2.193)

By subs.

45

𝑖𝑛𝑣𝛼𝑛𝑝2 =𝑠𝑛2

2𝑟2𝑣− 𝑡𝑎𝑛𝛼𝑛 + 𝛼𝑛

(2.194)

𝛼𝑛𝐿2 = 𝑡𝑎𝑛𝛼𝑛𝑤2 + 𝑖𝑛𝑣𝛼𝑛𝑝2

(2.195)

𝛼𝑛𝐿2 = 𝑡𝑎𝑛𝛼𝑛𝑤2 − 𝑡𝑎𝑛𝛼𝑛 + 𝛼𝑛 +𝑠𝑛2

2𝑟2𝑣

(2.196)

𝑟𝑛𝐿2 =𝑟𝑏2𝑣


(2.197)

Virtual Gear Root Evaluation for Internal Gear Pairs

Virtual gear root evaluation of the internal gears is outlined in this chapter. The

following equations are derived from Figure 2-21.

𝑅𝑓2𝑣 = √(√(𝑟𝑓2𝑣 − 𝐴2𝑣)2− 𝑟𝑏2𝑣

2 + 𝐴2𝑣)

2

+ 𝑟𝑏2𝑣2

(2.198)

𝛾2𝑣 = 𝑎𝑟𝑐𝑐𝑜𝑠 (𝑟𝑏2𝑣

𝑟𝑓2𝑣 − 𝐴2𝑣)

(2.199)

𝛼𝑅𝑓2𝑣= 𝑎𝑟𝑐𝑐𝑜𝑠 (

𝑟𝑏2𝑣

𝑅𝑓2𝑣)

(2.200)

𝑖𝑛𝑣𝛼𝑅𝑓2𝑣= 𝑡𝑎𝑛𝛼𝑅𝑓2𝑣

− 𝛼𝑅𝑓2𝑣

(2.201)

𝜃2𝑣 = 𝑎𝑟𝑐𝑠𝑖𝑛 (𝑟𝑏2𝑣

𝑟𝑓2𝑣 − 𝐴2𝑣)

(2.202)

𝑏2𝑣 = (𝑟𝑓2𝑣 − 𝐴2𝑣)𝑐𝑜𝑠 (𝜃𝑝2𝑣 − (𝑖𝑛𝑣𝛼𝑅𝑓2𝑣− 𝑖𝑛𝑣𝛼𝑛) − (𝛼𝑅𝑓2𝑣

− 𝛾2𝑣))

(2.203)

𝜃𝑥2𝑣 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝑎2𝑣

𝑏2𝑣)

(2.204)

The equation of the virtual gear root of the gear is expressed as:

𝑎2𝑣 + 𝐴2𝑣𝑠𝑖𝑛(𝜃2𝑣 + 𝜃𝑥2𝑣) − 𝑅𝑓2𝑣𝑠𝑖𝑛 (𝜃𝑝2𝑣 − (𝑖𝑛𝑣𝛼𝑅𝑓2𝑣− 𝑖𝑛𝑣𝛼𝑛)) = 0

(2.205)

This nonlinear equation is solved by implementing an iterative method. The equation

is solved by using fsolve in MATLAB in this study.

46

Figure 2-21 Normal plane view of virtual spur gear

Critical Section Determination for Internal Gear Pairs

The related equations for critical section evaluation of internal gear are outlined in this

chapter. As seen from Figure 2-22,



(

𝑎2𝑣

2 + 𝑏2𝑣2 + 𝑟𝑐𝑟𝑣2

2 − 𝐴2𝑣2


2

)

(2.206)

𝐿ℎ𝑣2 =𝑟𝑏2𝑣

cos (𝛼𝑛𝐿2)

(2.207)

ℎ𝑓2 = 𝑟𝑐𝑟𝑣2𝑐𝑜𝑠 (𝜋

𝑧2𝑣− 𝛼𝑐𝑟𝑣2) − 𝐿ℎ𝑣2

(2.208)

47



(2.209)

Figure 2-22 Critical section evaluation of virtual spur gear

Evaluation of the Critical Radius for Internal Gear Pairs

Evaluation of critical radius for internal gear is outlined in this section.

𝑟𝑐𝑟𝑣2(1) = 𝑅𝑓2𝑣 (2.210)

48

𝑟𝑐𝑟𝑣2(𝑖 + 1) = 𝑟𝑐𝑟𝑣2(𝑖) +(𝑟𝑓2𝑣 − 𝑅𝑓2𝑣)

𝑖𝑡𝑟

(2.211)

The consisted radii are substituted into the formulas of 𝛼𝑐𝑟𝑣2, ℎ𝑓2 and 𝑆𝐹2. Then, the

related 𝑌2, 𝐾𝑓2 values are obtained for each different 𝛼𝑐𝑟𝑣2, ℎ𝑓2 and 𝑆𝐹2. For each

different 𝑌2, 𝐾𝑓2 values, the different geometry factor values of the pinion ( 𝐽2 ) are

evaluated. Mathematically it is expressed as:

𝒇𝒐𝒓 𝒊 = 𝟏: 𝒊𝒕𝒓

𝛼𝑐𝑟𝑣2(𝑖) = arctan (𝑎2𝑣


(

𝑎2𝑣

2 + 𝑏2𝑣2 + (𝑟𝑐𝑟𝑣2(𝑖))

2− 𝐴2𝑣

2

2𝑟𝑐𝑟𝑣2(𝑖)√𝑎2𝑣2 + 𝑏2𝑣

2

)

ℎ𝑓2(𝑖) = 𝑟𝑐𝑟𝑣2(𝑖)𝑐𝑜𝑠 (𝜋

𝑧2𝑣− 𝛼𝑐𝑟𝑣2(𝑖)) − 𝐿ℎ𝑣2

𝑆𝐹2(𝑖) = 2𝑟𝑐𝑟𝑣2(𝑖)𝑠𝑖𝑛 (𝜋





6ℎ𝑓2(𝑖)



𝑆𝐹2(𝑖)]

𝐾𝑓2(𝑖) = 𝐻 + (𝑆𝐹2(𝑖)

𝐴2𝑣)

𝐿

(𝑆𝐹2(𝑖)

ℎ𝑓2(𝑖))

𝑀

𝐽2(𝑖) =𝑌2(𝑖)𝐶𝜓

𝐾𝑓2(𝑖)𝑚𝑁

𝒆𝒏𝒅

The minimum one of the 𝐽2 is taken as the geometry factor and implemented into the

bending stress equation.

49

2.1.2.3. Scuffing Evaluation

The term scuffing is defined as localized damage caused by solid-phase welding

between surfaces in relative motion. It is accompanied by transfer of metal from one

surface to another due to welding and subsequent tearing, and may occur in any highly

loaded contact where the oil film is too thin to adequately spate the surfaces. Scuffing

appears as a matte, rough finish due to the microscopic tearing at the surface. It occurs

most commonly at extreme end regions of the contact path or near points of single

tooth contact. Scuffing is also known generically as severe adhesive wear.

Scuffing risk is considered as the main criteria for scuffing elimination in optimization

process.

Table 2.6 Scuffing Risk [16]

Maximum contact temperature is given as:

𝜃𝐵 𝑚𝑎𝑥 = 𝜃𝑀 + 𝜃𝑓𝑙𝑚𝑎𝑥

(2.212)

Tooth temperature is given as:

𝜃𝑀 = 𝑘𝑠𝑢𝑚𝑝𝜃𝑜𝑖𝑙 + 0.56𝜃𝑓𝑙𝑚𝑎𝑥

(2.213)

where

𝑘𝑠𝑢𝑚𝑝 = 1.0 if splash lube; 1.2 if spray lube;

In the current study, temperature of the oil is considered as 100 ℃.

𝜃𝑜𝑖𝑙 = 100 ℃

Flash temperature is given as:

50

𝜃𝑓𝑙(𝑖)= 31.62𝐾𝜇𝑚(𝑖)

𝑋Γ(𝑖)𝑤𝑛

(𝑏𝐻(𝑖))0.5

|𝑣𝑟1(𝑖)− 𝑣𝑟2(𝑖)

|

𝐵𝑀1 (𝑣𝑟1(𝑖))0.5

+ 𝐵𝑀2 (𝑣𝑟2(𝑖))0.5

(2.214)

Mean coefficient of friction, 𝜇𝑚(𝑖) is given as:

𝜇𝑚(𝑖)= 𝜇𝑚𝑐𝑜𝑛𝑠𝑡

= 0.06𝐶𝑅𝑎𝑣𝑔𝑥

(2.215)

The surface roughness constant, 𝐶𝑅𝑎𝑣𝑔𝑥

, is limited to a maximum value of 3.0:

1.0 ≤ 𝐶𝑅𝑎𝑣𝑔𝑥=

1.13

1.13 − 𝑅𝑎𝑣𝑔𝑥

≤ 3.0

(2.216)

𝑅𝑎𝑣𝑔𝑥=

𝑅𝑎1𝑥+ 𝑅𝑎2𝑥

2

(2.217)

The thermal contact coefficient accounts for the influence of the material properties

of pinion and gear:

𝐵𝑀1 = (𝜆𝑀1𝜌𝑀1𝑐𝑚1)0.5

(2.218)

𝐵𝑀2 = (𝜆𝑀2𝜌𝑀2𝑐𝑚2)0.5

(2.219)

For martensitic steels the range of heat conductivity, 𝜆𝑀, is 41 to 52 𝑁/[𝑠 𝐾] and the

product of density times the specific heat per unit mass, 𝜌𝑚𝑐𝑚 is about

3.8 𝑁/[𝑚𝑚2𝐾], so that the use of the average value 𝐵𝑀 = 13.6 𝑁/[𝑚𝑚𝑠0.5𝐾] for

such steels will not introduce a large error when the thermal contact coefficient is

unknown.

Hertzian contact band is given as:

The semi-width of the rectangular contact band is given by:

𝑏𝐻(𝑖)= (

8𝑋Γ(𝑖)𝑤𝑛𝜌𝑛(𝑖)

𝜋𝐸𝑟)

0.5

(2.220)

𝐸𝑟 is reduced modulus of elasticity given by:

51

𝐸𝑟 = 2(1 − 𝑣1

2

𝐸1+

1 − 𝑣22

𝐸2)

−1

(2.221)

Evaluating the roll angles is necessary to implement the load sharing factor. Pinion

roll angles corresponding to the five specific points along the line of action shown in

Figure 2-23 are given by:

𝜉𝑗 =𝐶𝑖

𝑟𝑏1

(2.222)

where

𝑗 = 𝐴, 𝐵, 𝐶, 𝐷, 𝐸

𝑖 = 1,2,3,4,5

Profile Radii of Curvature

Figure 2-23 shows the transverse radii of curvature, 𝜌1(𝑖) and 𝜌2(𝑖)

, of the gear tooth

profiles at a general contact point defined by the roll angle 𝜉(𝑖), where (𝑖) is any point

on the line of action from 𝐴 to 𝐸.

Figure 2-23 Transverse relative radius of curvature for external gears [16]

52

𝜌1(𝑖)= 𝑟𝑏1𝜉(𝑖)

(2.223)

where

𝜉𝐴 ≤ 𝜉(𝑖) ≤ 𝜉𝐸

𝜌2(𝑖)= 𝐶6 ∓ 𝜌1(𝑖)

(2.224)

Transverse relative radius of curvature

𝜌𝑟(𝑖)=

𝜌1(𝑖)𝜌2(𝑖)

𝜌2(𝑖)± 𝜌1(𝑖)

(2.225)

Normal relative radius of curvature

𝜌𝑛(𝑖)=

𝜌𝑟(𝑖)


(2.226)

Figure 2-24 Transverse relative radius of curvature for internal gears

53

Gear Tooth Velocities and Loads

Rotational (angular) velocities

𝑤1 =𝜋𝑛1

30

(2.227)

𝑤2 =𝑤1

𝑚𝐺

(2.228)

Operating pitch line velocity

𝑣𝑡 =𝑤1𝑟𝑤1

1000

(2.229)

Rolling (tangential) velocities

𝑣𝑟1(𝑖)=

𝑤1𝜌1(𝑖)

1000

(2.230)

𝑣𝑟2(𝑖)=

𝑤2𝜌2(𝑖)

1000

(2.231)

Sliding velocity (absolute value)

𝑣𝑠(𝑖)= |𝑣𝑟1(𝑖)

− 𝑣𝑟2(𝑖)|

(2.232)

Entraining velocity (absolute value)

𝑣𝑒(𝑖)= |𝑣𝑟1(𝑖)

+ 𝑣𝑟2(𝑖)|

(2.233)

Nominal tangential load

(𝐹𝑡)𝑛𝑜𝑚 =1000𝑃

𝑣𝑡

(2.234)

Combined derating factor

𝐾𝐷 = 𝐾𝑜𝐾𝑚𝐾𝑣

(2.235)

where

𝐹𝑡 = (𝐹𝑡)𝑛𝑜𝑚𝐾𝐷

(2.236)

Normal operating load

54

𝐹𝑤𝑛 =𝐹𝑡

𝑐𝑜𝑠𝛼𝑤𝑛𝑐𝑜𝑠𝛽𝑤

(2.237)

Normal unit load

𝑤𝑛 =𝐹𝑤𝑛

𝐿𝑚𝑖𝑛

(2.238)

Load Sharing Factor

For unmodified tooth profiles

If there is no tip or root relief

For 𝜉𝐴 ≤ 𝜉(𝑖) < 𝜉𝐵

𝑋Γ(𝑖)=

1

3+

1

3(𝜉(𝑖) − 𝜉𝐴

𝜉𝐵 − 𝜉𝐴)

(2.239)

For 𝜉𝐵 ≤ 𝜉(𝑖) ≤ 𝜉𝐷

𝑋Γ(𝑖)= 1

(2.240)

For 𝜉𝐷 < 𝜉(𝑖) ≤ 𝜉𝐸

𝑋Γ(𝑖)=

1

3+

1

3(𝜉𝐸 − 𝜉(𝑖)

𝜉𝐸 − 𝜉𝐷)

(2.241)

Figure 2-25 Load sharing factor – unmodified profiles [16]

55

Evaluation of Maximum Flash Temperature

The roll angles are evaluated to find the maximum flash temperature.

𝒇𝒐𝒓 𝒊 = 𝟏: 𝟒𝟖

𝜉1 = 𝜉𝐴

𝒊𝒇 𝑖 <𝑖𝑡𝑝

4

𝜉𝑖+1 = 𝜉𝑖 +𝜉𝐵 − 𝜉𝐴

𝑖𝑡𝑝4

− 1

𝒊𝒇 𝑖 ≥𝑖𝑡𝑝

4 && 𝑖 < 2

𝑖𝑡𝑝

4

𝜉𝑖+1 = 𝜉𝑖 +𝜉𝐶 − 𝜉𝐵

𝑖𝑡𝑝4

𝒊𝒇 𝑖 ≥ 2𝑖𝑡𝑝

4 && 𝑖 < 3

𝑖𝑡𝑝

4

𝜉𝑖+1 = 𝜉𝑖 +𝜉𝐷 − 𝜉𝐶

𝑖𝑡𝑝4

𝒊𝒇 𝑖 ≥ 3𝑖𝑡𝑝

4 && 𝑖 < 𝑖𝑡𝑝

𝜉𝑖+1 = 𝜉𝑖 +𝜉𝐸 − 𝜉𝐷

𝑖𝑡𝑝4

𝒆𝒏𝒅

𝜌1(𝑖), 𝜌2(𝑖)

, 𝜌𝑟(𝑖), 𝜌𝑛(𝑖)

, 𝑣𝑟1(𝑖), 𝑣𝑟2(𝑖)

, 𝑋Γ(𝑖) and 𝑏𝐻(𝑖)

are obtained by implementing the all

different roll angles. Therefore, different flash temperatures values, 𝜃𝑓𝑙(𝑖) , for each

roll angle are obtained. The maximum one of the different flash temperature values is

56

implemented into the equation (2.213) to find the tooth temperature. Tooth

temperature is implemented into eq (2.212) to evaluate the contact temperature.

Speed parameter,

𝑈(𝑖) =𝜂𝑀𝑣𝑒(𝑖)

2𝐸𝑟𝜌𝑛(𝑖)

𝑥 10−6

(2.242)

Load parameter,

𝑊(𝑖) =𝑋Γ(𝑖)

𝑤𝑛

𝐸𝑟𝜌𝑛(𝑖)

(2.243)

Dimensionless central film thickness,

𝐻𝑐(𝑖)= 3.06

𝐺0.56𝑈(𝑖)0.69

𝑊(𝑖)0.10

(2.244)

The central film thickness at a given point is:

ℎ𝑐(𝑖)= 𝐻𝑐(𝑖)

𝜌𝑛(𝑖) 𝑥 103

(2.245)

The specific film thickness is:

𝜆2𝑏𝐻(𝑖)=

ℎ𝑐(𝑖)

𝜎0.8[

𝐿0.8

2𝑏𝐻(𝑖)

]

0.5

(2.246)

where

𝜎𝑥 = [𝑅𝑎1𝑥2 + 𝑅𝑎2𝑥

2 ]0.5

𝐿0.8 = 0.8 𝑚𝑚

57

CHAPTER 3

3. OPTIMIZATION METHODOLOGY

3.1. Root Clearance Criteria

Addendum and dedendum coefficient of the standard cutters are given in Table 3.1.

Coefficients are given for standard center distance.

Table 3.1 Addendum and dedendum coefficient of standard cutters

Standard Profil Type Dedendum

coefficient

Addendum

coefficient

ISO 53.2:1997 Profil A 1.25 1.00

ISO 53.2:1997 Profil B 1.25 1.00

ISO 53.2:1997 Profil C 1.25 1.00

ISO 53.2:1997 Profil D 1.40 1.00

DIN 867:1986 - 1.25 1.00

DIN 3972:1952 I - 1.1670 1.00

DIN 3972:1952 II - 1.25 1.00

DIN 58400:1984 (𝑚𝑛 0.1 − 0.6) 1.50 1.10

DIN 58400:1984 (𝑚𝑛 > 0.1 − 0.6) 1.35 1.00

DIN 58412:1984 (𝑚𝑛 0.1 − 0.6) 1.50 1.10

DIN 58412:1984 (𝑚𝑛 > 0.1 − 0.6) 1.35 1.10

DIN 58412:1987 - 1.25 1.00

DIN 867:1986 - 1.20 1.00

DIN 867:1986 - 1.16 1.00

DIN 867:1986 - 1.30 1.00

DIN 867:1986 - 1.40 1.00

As seen from Table 3.1, the standard cutters generally satisfy 0.25 ∙ 𝑚𝑡 root clearance.

The minimum standard root clearance is 0.16 ∙ 𝑚𝑡. The maximum standard root

clearance is 0.4 ∙ 𝑚𝑡. Therefore, the minimum root clearance is specified as 0.16 ∙ 𝑚𝑡

and the maximum root clearance is specified as 0.4 ∙ 𝑚𝑡

58

3.2. Top Land Thickness Criteria

Surface hardened gear teeth require adequate case depth to resist the subsurface shear

stresses developed by tooth contact loads and the tooth root fillet tensile stresses, but

depths must not be so great as to result in brittle teeth and high residual tensile stress

in the core.

Correlation between the normal diametral pitch and the effective case depth is given

in Figure 3-1.

Figure 3-1 Minimum effective case depth for carburized gears, ℎ𝑒 𝑚𝑖𝑛 [14]

For heavy case depth,

ℎ𝑒 𝑚𝑖𝑛 = 0.264693𝑃𝑛𝑑

−1.12481 (3.1)

59

The unit is converted into 𝑚𝑚 , the equation becomes

ℎ𝑒 𝑚𝑖𝑛 = 0.264693 (

25.4

𝑚𝑛)−1.12481

∙ 25.4

(3.2)

The tolerance of the carburizing depth can be taken as 0.25 𝑚𝑚. Therefore, the

maximum carburizing depth is:

ℎ𝑒 𝑚𝑎𝑥 = ℎ𝑒 𝑚𝑖𝑛 + 0.250 (3.3)

The relation between the top land thickness and the maximum effective case depth is

given as:

ℎ𝑒 𝑚𝑎𝑥 = 0.56𝑠𝑛𝑎

(3.4)

By equating the (3.3) and (3.4),

𝑠𝑛𝑎 𝑚𝑖𝑛 =

(0.264693 (25.4𝑚𝑛

)−1.12481

∙ 25.4 + 0.250)

0.56

(3.5)

3.3. Contact Stress Criteria

Load cycle formula is given as: [14]

𝑁 = 60𝐿𝑛𝑞 (3.6)

Aerospace gears are considered as resisted to 5000 ℎ𝑜𝑢𝑟𝑠 life. After 5000 ℎ𝑜𝑢𝑟𝑠 the

gears are inspected. If there exist no failure in gears, the gears are expected as having

infinite life. Therefore, the life is taken as 5000 ℎ𝑜𝑢𝑟𝑠 in this study. Since the single

gear pairs are considered in this study, 𝑞 is taken as unity.Allowable contact stress

number for steel gears is given in Table 3.2.

60

Table 3.2 Allowable contact stress number, 𝑠𝑎𝑐 , for steel gears [14]

Grade 3 carburized and hardened steel is considered in this study. Therefore, allowable

contact stress number is:

𝑠𝑎𝑐 = 275 𝑘𝑠𝑖 (3.7)

Contact strength stress cycle factor, 𝑍𝑁 , is associated with the lubrication regime in

AGMA 925. If the specific film thickness is greater than or equal to 1.0 indicates the

beginning of regime III and the end of regime II lubrication. Specific film thickness

between 0.4 and 1.0 indicates operation within regime II and specific film thickness

less than or equal to 0.4 indicates regime I.

Table 3.3 Stress cycle factor equations for regimes I, II and III [14]

61

Since the pinion has a higher rotational speed than the gear, number of load cycle of

the pinion is higher than the number of load cycle of the gear. Because of that, pinion

has a lower allowable contact stress number than the gear has. Therefore, considering

only the contact stress of the pinion is enough to evaluate the contact stress

elimination.

Permissible allowable contact stress number of the pinion is:

𝑠𝑎𝑐𝑝 = 𝑠𝑎𝑐

𝑍𝑁

𝐾𝑇

𝐶𝐻

𝐾𝑅

(3.8)

The case hardened aerospace gears have minimum 60 HRC surface hardness values.

Surface hardness values of the pinion and the gear are almost the same. In [2001-D04],

hardness ratio factor for case hardened pinions (48 HRC or harder) are run with

through hardened gears (180 to 400 HB) is given in Figure 3-2.

Figure 3-2 Hardness ratio factor, 𝐶𝐻 (through hardened) [14]

62

As seen from Figure 3-2, if the ratio of the hardness value of the pinion to hardness

ratio value of the gear is lower than 1.2, hardness ratio factor is taken as unity even

for the gear pairs which have a minimum 5 HRC hardness value difference between

the pinion and the gear. In this study, the gear pairs which have a small amount of

difference between the hardness value of the pinion and the hardness ratio of the gear

are considered. Therefore, it is suitable to consider the hardness ratio factor as unity.

The temperature factor, 𝐾𝑇, is generally taken as unity when gears operate with

temperatures of oil or gear blank not exceeding 250° 𝐹. In normal operating

conditions, temperature of oil or gear blank is not higher than 250° 𝐹 . Therefore, the

temperature factor is taken as unity.

The allowable stress numbers given in Table 3.2 and Table 3.5 are based upon a

statistical probability of one failure in 100. Therefore, reliability factor, 𝐾𝑅, is taken

as unity as given in Table 3.4.

Table 3.4 Reliability factors, 𝐾𝑅 [14]

The contact stress reserve factor is:

𝑅𝑐 =𝑠𝑎𝑐𝑝

𝑠𝑐

(3.9)

63

3.4. Bending Stress Criteria

Allowable bending stress number, 𝑠𝑎𝑡 , for steel gears is given in Table 3.5.

Table 3.5 Allowable bending stress number, 𝑠𝑎𝑡 , for steel gears [14]

Grade 3 carburized and hardened steel is considered in this study. Therefore, allowable

bending stress number is:

𝑠𝑎𝑡 = 75 𝑘𝑠𝑖 (3.10)

Bending strength stress cycle factor is given in Figure 3-3 . Bending strength stress

cycle factor is specified as:

𝑌𝑁 = 1.6831 ∙ 𝑁−0.0323

(3.11)

Permissible allowable bending stress number is: [14]

𝑠𝑎𝑡𝑝 =𝑠𝑎𝑡𝑌𝑁

𝐾𝑇𝐾𝑅

(3.12)

64

The bending stress reserve factor is:

𝑅𝑡 =𝑠𝑎𝑡𝑝

𝑠𝑡

(3.13)

Bending strength stress cycle factor is given in Figure 3-3 .

Figure 3-3 Bending strength stress cycle factor, 𝑌𝑁 [14]

65

Table 3.6 Minimum and maximum values of the constraint functions used in

optimization

Constraint Minimum Value Maximum Value

𝑐1 0.16 ∙ 𝑚𝑡 0.40 ∙ 𝑚𝑡

𝑐2 0.16 ∙ 𝑚𝑡 0.40 ∙ 𝑚𝑡

𝑠𝑛𝑎 (0.264693 (

25.4𝑚𝑛

)−1.12481

∙ 25.4 + 0.250)

0.56

𝑚𝑐 1

𝐼𝑐1 0.10 ∙ 𝑚𝑡

𝐼𝑐2 0.10 ∙ 𝑚𝑡

𝑇𝑖𝑓𝑓1 0.20 ∙ 𝑚𝑡

𝑇𝑖𝑓𝑓2 0.20 ∙ 𝑚𝑡

𝑅𝑐 1

𝑅𝑡1 1

𝑅𝑡2 1

𝑄 10

3.5. Optimization Steps

Optimization steps are given in this chapter.

3.5.1. Cross Combination of Normal Module, Helix Angle, Pinion Profile Shifting

Coefficient, Pressure Angle and Pinion Number of Teeth

The initial assignments of the normal module, helix angle, pinion profile shift

coefficient, pressure angle and pinion number of teeth are implemented.

Table 3.7 Initial assignments of the design variables

𝑚𝑛 = 𝑚𝑛𝑚𝑖𝑛: 𝑚𝑛𝑖𝑛𝑐

: 𝑚𝑛𝑚𝑎𝑥

𝛽 = 𝛽𝑚𝑖𝑛: 𝛽𝑖𝑛𝑐 ∶ 𝛽𝑚𝑎𝑥

𝑥1 = 𝑥1𝑚𝑖𝑛: 𝑥1𝑖𝑛𝑐

: 𝑥1𝑚𝑎𝑥

𝛼𝑛 = 𝛼𝑛𝑚𝑖𝑛: 𝛼𝑛𝑖𝑛𝑐

: 𝛼𝑛𝑚𝑎𝑥

𝑧1 = 𝑧1𝑚𝑖𝑛: 𝑧1𝑖𝑛𝑐

: 𝑧1𝑚𝑎𝑥

𝑧2 = 𝑚𝐺𝑧1

66

Cross combination of the 𝑚𝑛 , 𝛽 , 𝑥1 , 𝛼𝑛 and 𝑧1 is obtained. Cross combination matrix

is named as 𝐶𝑜𝑚𝑏𝑠1.

Table 3.8 First cross combination of the design variables

𝑚𝑛 = 𝐶𝑜𝑚𝑏𝑠1(: ,1)

𝛽 = 𝐶𝑜𝑚𝑏𝑠1(: ,2)

𝑥1 = 𝐶𝑜𝑚𝑏𝑠1(: ,3)

𝛼𝑛 = 𝐶𝑜𝑚𝑏𝑠1(: ,4)

𝑧1 = 𝐶𝑜𝑚𝑏𝑠1(: ,5)

𝑧2 = 𝐶𝑜𝑚𝑏𝑠1(: ,6) = 𝑚𝐺𝐶𝑜𝑚𝑏𝑠1(: ,5)

3.5.2. Evaluation of Pinion Addendum Radius and Gear Dedendum Radius

Addendum radius of the pinion is evaluated by using the following algorithm given in

Table 3.9.

Table 3.9 Evaluation of addendum radius of the pinion

𝑟𝑎1𝑝𝑟𝑒(1, : ) =

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑧1

2+ 𝑦1

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑓𝑜𝑟 𝑖 = 2 ∶ (𝑦2 − 𝑦1)

𝑖𝑛𝑐1+ 1

𝑟𝑎1𝑝𝑟𝑒(𝑖, : ) =

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑧1

2+ 𝑦1

𝑚𝑛

𝑐𝑜𝑠𝛽+ (𝑖𝑛𝑐1

𝑚𝑛

𝑐𝑜𝑠𝛽) (𝑖 − 1)

𝑒𝑛𝑑

𝑛𝑢𝑚1 =(𝑦2 − 𝑦1)

𝑖𝑛𝑐1+ 1

(3.14)

67

Therefore, 𝑛𝑢𝑚1 times addendum radii for external and internal pinion (for each row

of 𝐶𝑜𝑚𝑏𝑠1) are obtained by applying increment of 𝑖𝑛𝑐1.

Dedendum radii of the each gear set is evaluated by using the following algorithm

given in Table 3.10.

Table 3.10 Evaluation of dedendum radius of the gear

𝑟𝑓2𝑝𝑟𝑒(1, : ) =

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑧2

2+ 𝑦7

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑓𝑜𝑟 𝑖 = 2 ∶ (𝑦8 − 𝑦7)

𝑖𝑛𝑐4+ 1

𝑟𝑓2𝑝𝑟𝑒(𝑖, : ) =

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑧2

2+ 𝑦7

𝑚𝑛


𝑚𝑛

𝑐𝑜𝑠𝛽) (𝑖 − 1)

𝑒𝑛𝑑

𝑛𝑢𝑚4 =(𝑦8 − 𝑦7)

𝑖𝑛𝑐4+ 1

(3.15)

For example

𝐶𝑜𝑚𝑏𝑠1 = [ (𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2

]

(3.16)

If 𝑛𝑢𝑚1 is taken as ‘’3’’.

(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1

(𝑧1)1

2+ 𝑦1

(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1= 𝑟𝑎1𝑝𝑟𝑒

(1,1)

(3.17)

(𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2

(𝑧1)2

2+ 𝑦1

(𝑚𝑛)2


(1,2)

(3.18)

68

𝑟𝑎1𝑝𝑟𝑒

=

[

𝑟𝑎1𝑝𝑟𝑒(1,1) 𝑟𝑎1𝑝𝑟𝑒

(1,2)

𝑟𝑎1𝑝𝑟𝑒(1,1) + (𝑖𝑛𝑐1

(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1) ∙ 1 𝑟𝑎1𝑝𝑟𝑒

(1,2) + (𝑖𝑛𝑐1 (𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2) ∙ 1


(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1) ∙ 2 𝑟𝑎1𝑝𝑟𝑒

(1,2) + (𝑖𝑛𝑐1 (𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2) ∙ 2

]

(3.19)


(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1

(𝑧1)12

+ 𝑦7

(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1= 𝑟𝑓2𝑝𝑟𝑒

(1,1)

(3.20)

(𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2

(𝑧1)2

2+ 𝑦7

(𝑚𝑛)2


(1,2)

(3.21)

𝑟𝑓2𝑝𝑟𝑒

=

[

𝑟𝑓2𝑝𝑟𝑒(1,1) 𝑟𝑓2𝑝𝑟𝑒

(1,2)

𝑟𝑓2𝑝𝑟𝑒(1,1) + (𝑖𝑛𝑐4

(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1) ∙ 1 𝑟𝑓2𝑝𝑟𝑒

(1,2) + (𝑖𝑛𝑐4 (𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2) ∙ 1


(𝑚𝑛)1


(1,2) + (𝑖𝑛𝑐4 (𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2) ∙ 2

]

(3.22)

𝑟𝑓2𝑝𝑟𝑒

=

[


(1,2)


(𝑚𝑛)1


(1,2) + (𝑖𝑛𝑐4 (𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2) ∙ 1


(𝑚𝑛)1


(1,2) + (𝑖𝑛𝑐4 (𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2) ∙ 2

]

(3.23)

Cross combination of the 𝑟𝑎1𝑝𝑟𝑒 and 𝑟𝑓2𝑝𝑟𝑒

is implemented to obtain 𝑟𝑎1 and 𝑟𝑓2.

69

𝑟𝑎1 =

[ 𝑟𝑎1𝑝𝑟𝑒

(1,1)

𝑟𝑎1𝑝𝑟𝑒(2,1)
















𝑟𝑎1𝑝𝑟𝑒(3,2)]

𝑟𝑓2 =

[ 𝑟𝑓2𝑝𝑟𝑒

(1,1)

𝑟𝑓2𝑝𝑟𝑒(1,1)
















𝑟𝑓2𝑝𝑟𝑒(3,2)]

(3.24)

After obtaining the 𝑟𝑎1 and 𝑟𝑓2, combination of the 𝑚𝑛 , 𝛽 , 𝑥1 , 𝛼𝑛 , 𝑧1 , 𝑧2 , 𝑟𝑎1 and

𝑟𝑓2 is evaluated by expanding the each row of the 𝐶𝑜𝑚𝑏𝑠1 (𝑛𝑢𝑚1 ∙ 𝑛𝑢𝑚4) times.

Therefore second combination is expressed as:

70

𝐶𝑜𝑚𝑏𝑠2 =

[ (𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,1) 𝑟𝑓2(1,1)

(𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,2) 𝑟𝑓2(1,2)

(𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,3) 𝑟𝑓2(1,3)

(𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,4) 𝑟𝑓2(1,4)

(𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,5) 𝑟𝑓2(1,5)

(𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,6) 𝑟𝑓2(1,6)

(𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,7) 𝑟𝑓2(1,7)

(𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,8) 𝑟𝑓2(1,8)

(𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,9) 𝑟𝑓2(1,9)

(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,10) 𝑟𝑓2(1,10)

(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,11) 𝑟𝑓2(1,11)

(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,12) 𝑟𝑓2(1,12)

(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,13) 𝑟𝑓2(1,13)

(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,14) 𝑟𝑓2(1,14)

(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,15) 𝑟𝑓2(1,15)

(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,16) 𝑟𝑓2(1,16)

(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,17) 𝑟𝑓2(1,17)

(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,18) 𝑟𝑓2(1,18)]

(3.25)

Operating center distance for external gear pair is:

𝑎𝑤 =𝑚𝑛

𝑐𝑜𝑠𝛽

(𝑧1 + 𝑧2)

2

(3.26)

Operating center distance for internal gear pair is:

𝑎𝑤 =𝑚𝑛

𝑐𝑜𝑠𝛽

(𝑧2 − 𝑧1)

2

(3.27)

3.5.3. Gear Root Clearance Elimination

The gear root clearance elimination is implemented by applying the following

algorithm given in Table 3.11.

Table 3.11 Gear root clearance elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑐2) 𝑖𝑓 𝑐2(𝑖) < 0.17𝑚𝑡 || 𝑐2(𝑖) > 0.35𝑚𝑡 𝐶𝑜𝑚𝑏𝑠2(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

71

3.5.4. Pinion Top Land Thickness Elimination

Pinion top land thickness elimination is implemented by applying the following


Table 3.12 Pinion top land thickness elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑠𝑛𝑎𝑛1) 𝑖𝑓 𝑠𝑛𝑎𝑛1(𝑖) < 𝑠𝑛𝑎𝑛1 𝑚𝑖𝑛 𝐶𝑜𝑚𝑏𝑠2(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

Therefore, the gear pair sets which do not satisfy the pinion top land criteria is

eliminated.

3.5.5. Evaluation of Gear Addendum Radius and Pinion Dedendum Radius

After pinion top land thickness elimination is completed, evaluation of 𝑟𝑎2 and 𝑟𝑓1 is

done. Addendum radius of the gear is evaluated by using the following algorithm

given in Table 3.13 .

Table 3.13 Evaluation of addendum radius of the gear

𝑟𝑎2𝑝𝑟𝑒(1, : ) =

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑧2

2+ 𝑦3

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑓𝑜𝑟 𝑖 = 2 ∶ (𝑦4 − 𝑦3)

𝑖𝑛𝑐2+ 1

𝑟𝑎1𝑝𝑟𝑒(𝑖, : ) =

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑧2

2+ 𝑦3

𝑚𝑛


𝑚𝑛

𝑐𝑜𝑠𝛽) (𝑖 − 1)

𝑒𝑛𝑑

𝑛𝑢𝑚2 =(𝑦4 − 𝑦3)

𝑖𝑛𝑐2+ 1

(3.28)

72

Therefore, 𝑛𝑢𝑚2 times addendum radii for external and internal gear (for each row of

𝐶𝑜𝑚𝑏𝑠1) are obtained by applying increment of 𝑖𝑛𝑐2.

Pinion dedendum radii of the each gear set is evaluated by using the following


Table 3.14 Evaluation of dedendum radius of the pinion

𝑟𝑓1𝑝𝑟𝑒(1, : ) =

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑧1

2+ 𝑦6

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑓𝑜𝑟 𝑖 = 2 ∶ (𝑦6 − 𝑦5)

𝑖𝑛𝑐3+ 1

𝑟𝑓1𝑝𝑟𝑒(𝑖, : ) =

𝑚𝑛

𝑐𝑜𝑠𝛽

𝑧1

2+ 𝑦6

𝑚𝑛


𝑚𝑛

𝑐𝑜𝑠𝛽) (𝑖 − 1)

𝑒𝑛𝑑

𝑛𝑢𝑚3 =(𝑦6 − 𝑦5)

𝑖𝑛𝑐3+ 1

(3.29)

Let’s assume after the gear root clearance elimination and pinion top land thickness

elimination is completed, 𝐶𝑜𝑚𝑏𝑠2 becomes:

𝐶𝑜𝑚𝑏𝑠2 = [(𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,9) 𝑟𝑓2(1,9)

(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,6) 𝑟𝑓2(1,6)]

(3.30)


(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1

(𝑧2)1

2+ 𝑦3

(𝑚𝑛)1


(1,1)

(3.31)

(𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2

(𝑧2)2

2+ 𝑦3

(𝑚𝑛)2


(1,2)

(3.32)

𝑟𝑎2𝑝𝑟𝑒=

[

𝑟𝑎2𝑝𝑟𝑒(1,1) 𝑟𝑎2𝑝𝑟𝑒

(1,2)


(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1

) ∙ 1 𝑟𝑎2𝑝𝑟𝑒(1,2) + (𝑖𝑛𝑐2

(𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2

) ∙ 1


(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1

) ∙ 2 𝑟𝑎2𝑝𝑟𝑒(1,2) + (𝑖𝑛𝑐2

(𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2

) ∙ 2]

(3.33)

73


(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1

(𝑧1)12

+ 𝑦5

(𝑚𝑛)1


(1,1)

(3.34)

(𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2

(𝑧1)2

2+ 𝑦5

(𝑚𝑛)2


(1,2)

(3.35)

𝑟𝑓1𝑝𝑟𝑒=

[


(1,2)


(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1

) ∙ 1 𝑟𝑓1𝑝𝑟𝑒(1,2) + (𝑖𝑛𝑐3

(𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2

) ∙ 1


(𝑚𝑛)1

𝑐𝑜𝑠(𝛽)1

) ∙ 2 𝑟𝑓1𝑝𝑟𝑒(1,2) + (𝑖𝑛𝑐3

(𝑚𝑛)2

𝑐𝑜𝑠(𝛽)2

) ∙ 2]

(3.36)

Cross combination of the 𝑟𝑎2𝑝𝑟𝑒 and 𝑟𝑓1𝑝𝑟𝑒

is implemented to obtain 𝑟𝑎2 and 𝑟𝑓1.

𝑟𝑎2 =

[ 𝑟𝑎2𝑝𝑟𝑒

(1,1)

















𝑟𝑎2𝑝𝑟𝑒(3,2)]

𝑟𝑓1 =

[ 𝑟𝑓1𝑝𝑟𝑒

(1,1)

















𝑟𝑓1𝑝𝑟𝑒(3,2)]

(3.37)

74

After obtaining the 𝑟𝑎2 and 𝑟𝑓1, combination of the 𝑚𝑛 , 𝛽 , 𝑥1 , 𝛼𝑛 , 𝑧1 , 𝑧2 , 𝑟𝑎1 , 𝑟𝑓2 ,

𝑟𝑎2 and 𝑟𝑓1 is evaluated by expanding the each row of the 𝐶𝑜𝑚𝑏𝑠2 (𝑛𝑢𝑚2 ∙ 𝑛𝑢𝑚3)

times. Therefore third combination is expressed as:

𝐶𝑜𝑚𝑏𝑠3 =

[ (𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,6) 𝑟𝑓2(1,6) 𝑟𝑎2(1,1) 𝑟𝑓1(1,1)

(𝑚𝑛)1 (𝛽)1 (𝑥1)1 (𝛼𝑛)1 (𝑧1)1 (𝑧2)1 𝑟𝑎1(1,6) 𝑟𝑓2(1,6) 𝑟𝑎2(1,2) 𝑟𝑓1(1,2)
















(𝑚𝑛)2 (𝛽)2 (𝑥1)2 (𝛼𝑛)2 (𝑧1)2 (𝑧2)2 𝑟𝑎1(1,9) 𝑟𝑓2(1,6) 𝑟𝑎2(1,18) 𝑟𝑓1(1,18)]

(3.38)

3.5.6. Pinion Root Clearance Elimination

The gear root clearance elimination is implemented by applying the following


Table 3.15 Pinion root clearance elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑐1) 𝑖𝑓 𝑐1(𝑖) < 0.17𝑚𝑡 || 𝑐1(𝑖) > 0.35𝑚𝑡 𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

3.5.7. Gear Top Land Thickness Elimination

Gear top land thickness elimination is implemented by applying the following


75

Table 3.16 Gear top land thickness elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑠𝑛𝑎𝑛2)

𝑖𝑓 𝑠𝑛𝑎𝑛2(𝑖) < 𝑠𝑛𝑎𝑛2 𝑚𝑖𝑛

𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ]

𝑒𝑛𝑑

𝑒𝑛𝑑

3.5.8. Contact Ratio Elimination

The contact ratio is implemented by applying the following algorithm given in Table

3.17.

Table 3.17 Contact ratio elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑚𝑐)

𝑖𝑓 𝑚𝑐(𝑖) < 1

𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ]

𝑒𝑛𝑑

𝑒𝑛𝑑

3.5.9. Involute Clearance Elimination

Pinion involute clearance elimination is implemented by applying the following


Table 3.18 Pinion involute clearance elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝐼𝑐1) 𝑖𝑓 𝐼𝑐1(𝑖) > 0.1𝑚𝑡 𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

76

Gear involute clearance elimination is implemented by applying the following


Table 3.19 Gear involute clearance elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝐼𝑐2) 𝑖𝑓 𝐼𝑐2(𝑖) > 0.1𝑚𝑡 𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

3.5.10. Tiff Clearance Elimination

Pinion tiff clearance elimination is implemented by applying the following algorithm


Table 3.20 Pinion tiff clearance elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑇𝑖𝑓𝑓1) 𝑖𝑓 𝑇𝑖𝑓𝑓1(𝑖) < 0.2𝑚𝑡 𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

Gear tiff clearance elimination is implemented by applying the following algorithm


Table 3.21 Gear tiff clearance elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑇𝑖𝑓𝑓2) 𝑖𝑓 𝑇𝑖𝑓𝑓2(𝑖) < 0.2𝑚𝑡 𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

77

3.5.11. Contact Stress Elimination

Contact stress elimination is implemented by applying the following algorithm given

in Table 3.22.

Table 3.22 Contact stress elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑅𝑐) 𝑖𝑓 𝑅𝑐(𝑖) < 1 𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

3.5.12. Bending Stress Elimination

Bending stress elimination is implemented by applying the following algorithm given

in Table 3.23 and Table 3.24.

Table 3.23 Pinion bending stress elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑅𝑡1) 𝑖𝑓 𝑅𝑡1(𝑖) < 1 𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

Table 3.24 Gear bending stress elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑅𝑡2) 𝑖𝑓 𝑅𝑡2(𝑖) < 1 𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

78

3.5.13. Scuffing Elimination

Scuffing elimination is implemented by applying the following algorithm given in

Table 3.25.

Table 3.25 Scuffing elimination algorithm

𝑓𝑜𝑟 𝑖 = 1: 𝑠𝑖𝑧𝑒(𝑄) 𝑖𝑓 𝑄(𝑖) < 1 𝐶𝑜𝑚𝑏𝑠3(𝑖, : ) = [ ] 𝑒𝑛𝑑 𝑒𝑛𝑑

After all the eliminations are implemented, the gear pair which has the lowest

operating center distance is specified as the optimum gear pair among the remaining

gear pairs.

79

CHAPTER 4

4. SENSITIVITY ANALYSIS AND CASE STUDIES

4.1. Sensitivity Analysis and Case Studies of External Gear Pair and Internal

Gear Pair

Sensitivity analysis of external and internal gear pairs are implemented in this section.

Relatively high and low speed analyses are conducted to observe the speed effect on

the sensitivities of the design variables. The main aim in this section is to observe the

effect of the change of the increment of design variables on objective function. The

required increment values for each design variable are determined in this section.

4.1.1. Sensitivity Analysis of External Gear Pair

The external gear pairs are examined in the case studies from 1 to 10. The rotational

speed of the pinion is taken as 4000 𝑟𝑝𝑚 in the cases from 1 to 5. The pinion rotational

speed is taken as 1500 𝑟𝑝𝑚 in the cases from 6 to 10. Difference between the pinion

rotational speed is applied to observe the relatively high and low speeds on the

sensitivity of the design variables.

Table 4.1 gives the pinion and gear rotational speeds and the power input in the cases

from 1 to 5. Table 4.2 gives the upper and lower bounds and the increment values of

the pinion and gear addendum and dedendum radii.

Table 4.1 Speed and power values for the analysis from Case 1 to Case 5

Parameter Value

𝑛1 4000 𝑟𝑝𝑚

𝑛2 1017.81 𝑟𝑝𝑚

𝑃𝑜𝑤𝑒𝑟 1700 𝑘𝑊

80


dedendum radii for the analysis from Case 1 to Case 5

𝒓𝒂𝟏 𝒚𝟏 𝒊𝒏𝒄𝟏 𝒚𝟐

0.8 0.1 1.20

𝒓𝒂𝟐 𝒚𝟑 𝒊𝒏𝒄𝟐 𝒚𝟒

0.8 0.1 1.20

𝒓𝒇𝟏 𝒚𝟓 𝒊𝒏𝒄𝟑 𝒚𝟔

−1.5 0.1 −1.10

𝒓𝒇𝟐 𝒚𝟖 𝒊𝒏𝒄𝟒 𝒚𝟕

−1.5 0.1 −1.10

Initial assignments of the design variables are given in Table 4.3. In the first case,

pinion number of teeth is started from 10 and it is ended at 50 by increasing its value

1. The module is started from 2 𝑚𝑚 and it is ended at 7.7 𝑚𝑚 by increasing its value

0.3 𝑚𝑚. The pressure angle is started from 15° and it is ended at 35° by increasing its

value 1°. The pinion profile shifting coefficient is started from −0.6 and it is ended at

0.6 by increasing its value 0.2. The helix angle is started from 5° and it is ended at 21°

by increasing its value 2°. In the second cases, all the design parameters are kept as

the same as they are in the first case except the parameter of the module. The module

values are taken as constant values to observe the effect of the increment value of the

module on objective function. In the third cases, all the design parameters are kept as

the same as they are in the first case except the parameter of the helix angle. The helix

angle values are taken as constant values to observe the effect of the increment value

of the helix angle on objective function. In the fourth cases, all the design parameters

are kept as the same as they are in the first case except the parameter of the pinion

profile shifting coefficient. The pinion profile shifting coefficient values are taken as

constant values to observe the effect of the increment value of the pinion profile

shifting coefficient on objective function. In the fifth cases, all the design parameters

are kept as the same as they are in the first case except the parameter of the pressure

angle. The pressure angle values are taken as constant values to observe the effect of

the increment value of the pressure angle on objective function.

81

Table 4.3 Input parameters of the studies from Case1 to Case 5

Case 𝒛𝟏 𝒎𝒏 (𝒎𝒎) 𝜶𝒏(𝒅𝒆𝒈) 𝜷(𝒅𝒆𝒈) 𝒙𝟏

1 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

2 A 10 ∶ 1 ∶ 50 2.9 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

2 B 10 ∶ 1 ∶ 50 3.0 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

2 C 10 ∶ 1 ∶ 50 3.1 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

2 D 10 ∶ 1 ∶ 50 3.2 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

2 E 10 ∶ 1 ∶ 50 3.3 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

2 F 10 ∶ 1 ∶ 50 3.4 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

2 G 10 ∶ 1 ∶ 50 3.5 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

2 H 10 ∶ 1 ∶ 50 3.6 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

3 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 10 −0.6 ∶ 0.2 ∶ 0.6

3 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 11 −0.6 ∶ 0.2 ∶ 0.6

3 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 12 −0.6 ∶ 0.2 ∶ 0.6

3 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 13 −0.6 ∶ 0.2 ∶ 0.6

3 E 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 14 −0.6 ∶ 0.2 ∶ 0.6

3 F 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 15 −0.6 ∶ 0.2 ∶ 0.6

3 G 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 16 −0.6 ∶ 0.2 ∶ 0.6

3 H 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 17 −0.6 ∶ 0.2 ∶ 0.6

4 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.2

4 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.1

4 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0

4 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0.1

4 E 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0.2

4 F 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0.3

5 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 21 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

5 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 21.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

5 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 22 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

5 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 22.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

5 E 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 23 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

5 F 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 23.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

5 G 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 24 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

5 H 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 24.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

82

Table 4.4 Optimized design variables (Case1 – Case 5)

𝒛𝟏 𝒛𝟐 𝒎𝒏

𝜶𝒏

𝜷

𝒙𝟏 𝒓𝒂𝟏

𝒓𝒂𝟐

𝒓𝒇𝟏

𝒓𝒇𝟐

𝒂𝒘

Case - - 𝑚𝑚 𝑑𝑒𝑔 𝑑𝑒𝑔 - 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚

1 37 145 3.2 23 13 0.2 64.698 241.058 56.816 233.176 298.860

2 A 41 161 2.9 24 19 0.2 66.249 249.355 59.502 242.608 309.777

2 B 40 157 3.0 26 15 0.2 65.533 246.292 58.390 239.149 305.924

2 C 37 145 3.1 22 21 0.2 65.415 243.728 57.446 235.759 302.170

2 D 37 145 3.2 23 13 0.2 64.698 241.058 56.816 233.176 298.860

2 E 37 145 3.3 23 7 0.2 65.498 244.039 57.519 236.060 302.555

2 F 35 138 3.4 23 15 0.2 65.823 246.044 57.023 237.560 304.475

2 G 35 138 3.5 23 13 0.2 67.172 251.085 58.551 242.464 310.714

2 H 35 138 3.6 24 9 0.2 68.159 254.777 59.047 246.029 315.282

3 A 44 173 2.9 22 10 0.2 68.318 257.665 60.956 250.303 319.504

3 B 38 149 3.2 24 11 0.2 65.850 245.796 58.026 237.972 304.800

3 C 38 149 3.2 23 12 0.2 66.084 246.670 57.905 238.819 305.884

3 D 37 145 3.2 23 13 0.2 64.698 241.058 56.816 233.176 298.860

3 E 37 145 3.2 23 14 0.2 64.970 242.071 57.055 234.155 300.115

3 F 37 145 3.2 23 15 0.2 65.264 243.166 57.313 235.215 301.472

3 G 37 145 3.2 23 16 0.2 65.580 244.346 57.591 236.356 302.935

3 H 37 145 3.2 22 17 0.2 65.920 245.612 57.890 237.581 304.505

4 A 50 197 2.6 24 21 -0.2 71.852 277.384 65.725 270.978 343.945

4 B 40 157 3.2 25 7 -0.1 67.382 256.633 59.967 249.218 317.567

4 C 38 149 3.2 24 13 0 65.683 247.955 57.801 240.073 307.070

4 D 35 138 3.2 24 21 0.1 63.412 239.594 55.871 231.710 296.493

4 E 37 145 3.2 23 13 0.2 64.698 241.058 56.816 233.176 298.860

4 F 37 145 3.2 24 15 0.3 65.264 242.834 57.644 235.215 301.472

5 A 41 161 2.9 21 21 0.2 67.407 253.165 59.641 245.399 313.738

5 B 37 145 3.2 21.5 21 0.2 67.525 251.591 58.956 243.364 311.917

5 C 37 145 3.2 22 17 0.2 65.920 245.612 57.890 237.581 304.505

5 D 37 145 3.2 22.5 15 0.2 65.264 243.166 57.313 235.215 301.472

5 E 37 145 3.2 23 13 0.2 64.698 241.058 56.816 233.176 298.860

5 F 37 145 3.2 23.5 13 0.2 64.698 241.058 56.488 233.176 298.860

5 G 38 149 3.2 24 11 0.2 65.850 245.796 58.026 237.972 304.800

5 H 38 149 3.2 24.5 13 0 65.683 247.955 57.801 240.073 307.070

83

Table 4.4 and Table 4.5 gives the pinion and gear rotational speeds and the power

input in the cases from 1 to 6. Table 4.6 gives the upper and lower bounds and the

increment values of the pinion and gear addendum and dedendum radii.


Parameter Value

𝑛1 1500 𝑟𝑝𝑚

𝑛2 381.679 𝑟𝑝𝑚



dedendum radii for the analysis form Case 6 to Case 10


0.8 0.1 1.20


0.8 0.1 1.20


−1.5 0.1 −1.10


−1.5 0.1 −1.10

Initial assignments of the design variables are given in Table 4.7. In the sixth case,

pinion number of teeth is started from 1 and it is ended at 50 by increasing its value 1.

The module is started from 2 𝑚𝑚 and it is ended at 7.7 𝑚𝑚 by increasing its value




by increasing its number at 2°.

In the seventh cases, all the design parameters are kept as the same as they are in the

sixth case except the parameter of the module. The module values are taken as constant

values to observe the effect of the increment value of the module on objective function.

84

In the eight cases, all the design parameters are kept as the same as they are in the

sixth case except the parameter of the helix angle. The helix angle values are taken as

constant values to observe the effect of the increment value of the helix angle on

objective function. In the ninth cases, all the design parameters are kept as the same

as they are in the sixth case except the parameter of the pinion profile shifting

coefficient. The pinion profile shifting coefficient values are taken as constant values

to observe the effect of the increment value of the pinion profile shifting coefficient

on objective function. In the tenth cases, all the design parameters are kept as the same

as they are in the sixth case except the parameter of the pressure angle. The pressure


of the pressure angle on objective function.

85

Table 4.7 Input parameters of the studies from Case 6 to Case 10


6 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

7 A 10 ∶ 1 ∶ 50 4.1 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

7 B 10 ∶ 1 ∶ 50 4.2 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

7 C 10 ∶ 1 ∶ 50 4.3 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

7 D 10 ∶ 1 ∶ 50 4.4 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

7 E 10 ∶ 1 ∶ 50 4.5 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

7 F 10 ∶ 1 ∶ 50 4.6 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

7 G 10 ∶ 1 ∶ 50 4.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

7 H 10 ∶ 1 ∶ 50 4.8 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

8 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 18 −0.6 ∶ 0.2 ∶ 0.6

8 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 19 −0.6 ∶ 0.2 ∶ 0.6

8 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 20 −0.6 ∶ 0.2 ∶ 0.6

8 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 21 −0.6 ∶ 0.2 ∶ 0.6

8 E 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 22 −0.6 ∶ 0.2 ∶ 0.6

8 F 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 23 −0.6 ∶ 0.2 ∶ 0.6

8 G 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 24 −0.6 ∶ 0.2 ∶ 0.6

8 H 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 25 −0.6 ∶ 0.2 ∶ 0.6

9 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.1

9 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0

9 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0.1

9 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0.2

9 E 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0.3

10 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 22 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

10 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 22.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

10 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 23 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

10 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 23.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

10 E 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 24 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

10 F 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 24.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

10 G 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 25 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

10 H 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 25.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

86

Table 4.8 Optimized design variables (Case6 – Case10)


𝜶𝒏

𝜷


𝒓𝒂𝟐

𝒓𝒇𝟏

𝒓𝒇𝟐

𝒂𝒘


6 35 138 4.4 22 21 0.2 88.134 329.913 75.880 318.130 407.678

7 A 40 157 4.1 23 17 0.2 90.892 340.414 80.602 330.125 422.303

7 B 38 149 4.2 22 19 0.2 89.729 335.372 78.179 324.267 415.328

7 C 38 149 4.3 24 13 0.2 89.145 332.748 78.553 322.157 412.626

7 D 35 138 4.4 22 21 0.2 88.134 329.913 75.880 318.130 407.678

7 E 35 138 4.5 23 19 0.2 88.999 332.675 77.576 321.252 411.679

7 F 35 138 4.6 24 15 0.2 89.054 332.883 77.149 321.453 411.936

7 G 33 130 4.7 23 21 0.2 89.109 331.766 76.523 319.683 410.302

7 H 33 130 4.8 23 21 0.2 91.004 338.825 78.151 326.485 419.032

8 A 37 145 4.4 24 18 0 90.215 340.506 79.112 328.939 421.005

8 B 37 145 4.4 23 19 0.2 91.675 341.569 80.041 330.401 423.471

8 C 37 145 4.4 22 10 0.2 92.243 344.155 80.537 332.449 426.097

8 D 35 138 4.4 22 21 0.2 88.134 329.913 75.880 318.130 407.678

8 E 35 138 4.4 22 22 0.2 88.742 332.189 76.403 320.325 410.490

8 F 37 145 4.4 23 23 0 93.210 351.807 81.738 339.857 434.979

8 G 38 149 4.1 25 24 0.2 90.209 337.947 79.887 327.625 419.629

8 H 35 138 4.4 21 25 0.2 90.786 339.840 78.649 327.703 419.946

9 A 38 149 4.4 26 13 -0.1 89.863 341.390 79.477 331.004 422.222

9 B 37 145 4.4 24 17 0 89.720 338.637 78.678 327.594 418.695

9 C 35 138 4.4 23 21 0.1 87.663 329.913 76.351 318.130 407.678

9 D 35 138 4.4 22 21 0.2 88.134 329.913 75.880 318.130 407.678

9 E 43 169 3.8 24 21 0.3 92.397 347.201 82.628 337.839 431.457

10 A 35 138 4.4 22 21 0.2 88.134 329.913 75.880 318.130 407.678

10 B 37 145 4.4 22.5 17 0.2 90.641 338.177 79.138 326.674 418.695

10 C 35 138 4.4 23 21 0.2 88.134 329.441 76.351 318.130 407.678

10 D 37 145 4.4 23.5 13 0.2 88.960 331.455 77.671 320.617 410.932

10 E 37 145 4.4 24 13 0.2 88.960 331.455 77.671 320.617 410.932

10 F 37 145 4.4 24.5 13 0.2 88.960 331.455 77.671 320.617 410.932

10 G 38 149 4.4 25 7 0.2 89.547 334.251 78.465 323.612 414.490

10 H 35 138 4.7 25.5 15 0 90.017 340.606 78.826 329.415 420.892

87

The optimized results of the cases from 1 to 5 are given in Table 4.8. It is concluded

that the optimized center distance is 298.860 𝑚𝑚 in the first case with the module of

3.2 mm. When the Case 2A is considered it is seen that the optimized center distance

is 309.777 𝑚𝑚 with the module of 2.9 𝑚𝑚. Sensitivity of the Case 2A is considered

as follow:

𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 =(309.777 𝑚𝑚 − 298.860 𝑚𝑚)

(298.860 𝑚𝑚)∗ 100

The sensitivity calculation is conducted as given above for all other points in each case

study. The module sensitivity and the center distance relation at 4000 rpm and 1500

rpm are given in Figure 4-1 and Figure 4-2, respectively. The optimum center distance

and the optimum module value are obtained as 298.86 𝑚𝑚 and 3.2 mm respectively

in the first case. It is seen that the optimum center distances are 302.17 𝑚𝑚 and

302.555 𝑚𝑚 when the module values are adjusted to 3.1 𝑚𝑚 and 3.2 𝑚𝑚,

respectively. Therefore, changing the module value 0.1 𝑚𝑚 around the actual

optimum module deviates the center distance value 3.70 𝑚𝑚 at most. The sensitivity

of the module parameter is 1.23 % at most 0.1 𝑚𝑚 around the actual optimum module

value. When the 0.1 𝑚𝑚 increment of the module is considered sensitivity of the

module is always in a decreasing manner if the module values are moved away from

the optimum module point. The optimum center distance and the optimum module

value are obtained as 407.678 𝑚𝑚 and 4.4 mm respectively in the sixth case. It is

seen that the optimum center distances are 412.626 𝑚𝑚 and 411.679 𝑚𝑚 when the

module values are adjusted to 4.3 𝑚𝑚 and 4.5 𝑚𝑚, respectively. Therefore, changing

the module value 0.1 𝑚𝑚 around the actual optimum module deviates the center

distance value 4.948 𝑚𝑚 at most. The sensitivity of the module parameter is 1.21 %

at most 0.1 𝑚𝑚 around the actual optimum module value. When the 0.1 𝑚𝑚

increment of the module is considered sensitivity of the module is always in a

decreasing manner if the module values are moved away from the optimum module

88

point. The increment of the module value is specified as 0.1 𝑚𝑚 which gives a

sensitivity value of 1.23 % at most.


relatively higher pinion rotational speed (4000 rpm)


relatively lower pinion rotational speed (1500 rpm)

-0,05

0,45

0,95

1,45

1,95

2,45

2,95

3,45

3,95

4,45

4,95

5,45

5,95

297

299

301

303

305

307

309

311

313

315

2,8 2,9 3 3,1 3,2 3,3 3,4 3,5 3,6 3,7

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)

Module (mm)

Center Distance Sensitivity

-0,05

0,45

0,95

1,45

1,95

2,45

2,95

3,45

3,95

406

408

410

412

414

416

418

420

422

424

4 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 4,9

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)

Module (mm)


89

The helix angle sensitivity and the center distance relation at 4000 rpm and 1500 rpm

are given in Figure 4-3 and Figure 4-4, respectively. The optimum center distance and

the optimum helix angle value are obtained as 298.86 𝑚𝑚 and 13° respectively in the

first case. It is seen that the optimum center distances are 305.884 𝑚𝑚 and

300. 115 𝑚𝑚 when the helix angle values are adjusted to 12° and 14°, respectively.

Therefore, changing the helix angle value 1° around the actual optimum helix angle

deviates the center distance value 7.02 𝑚𝑚 at most. The sensitivity of the helix angle

parameter is 2.35 % at most 1° around the actual optimum helix angle value. When

the 1° increment of the helix angle is considered sensitivity of the helix angle in a

decreasing manner if the helix angle values are moved away from the optimum helix

angle point.

The optimum center distance and the optimum helix angle value are obtained as

407.678 𝑚𝑚 and 21° respectively in the sixth case. It is seen that the optimum center

distances are 426.097 𝑚𝑚 and 410.49 𝑚𝑚 when the helix angle values are adjusted

to 20° and 22°, respectively. Therefore, changing the helix angle value 1° around the

actual optimum helix angle deviates the center distance value 18.419 𝑚𝑚 at most.

The sensitivity of the helix angle parameter is 4.52 % at most 1°around the actual

optimum helix angle value. When the 1° increment of the helix angle is considered

sensitivity of the module is not always in a decreasing manner if the helix angle values

are moved away from the optimum helix angle point. The sensitivity value does not

show a consistent behavior as opposite to obtained in 4000 𝑟𝑝𝑚. Therefore, when the

rotational speed of the pinion is decreased the helix angle does not keep its consistent

behavior any more. Although this inconsistency, the sensitivity of the helix angle is

still below the 5 % 1°around the actual optimum helix angle value. The increment of

the helix angle value is specified as 1° which gives a sensitivity value of 4.52 % at

most.

90

Figure 4-3 Helix angle sensitivity and center distance relation of external gear pair

at relatively higher pinion rotational speed (4000 rpm)

Figure 4-4 Helix angle sensitivity and center distance relation of external gear pair

at relatively lower pinion rotational speed (1500 rpm)

9 10 11 12 13 14 15 16 17 18

-0,5

0,5

1,5

2,5

3,5

4,5

5,5

6,5

7,5

295

300

305

310

315

320

Helix Angle (degree)

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)


-1

0

1

2

3

4

5

6

7

406

409

412

415

418

421

424

427

430

433

436

17 18 19 20 21 22 23 24 25 26

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)

Helix Angle


91

The pinion profile shifting coefficient sensitivity and the center distance relation at

4000 rpm and 1500 rpm are given in Figure 4-5 and Figure 4-6, respectively. The

optimum center distance and the optimum profile shifting coefficient are obtained as

298.86 𝑚𝑚 and 0.2 respectively in the first case. It is seen that the optimum center

distances are 296.493 𝑚𝑚 and 301.472 𝑚𝑚 when the pinion profile shifting

coefficient values are adjusted to 0.1 and 0.3, respectively. Therefore, changing the

pinion profile shifting coefficient value 0.1 around the actual optimum profile shifting

coefficient deviates the center distance value 2.61 𝑚𝑚 at most. The sensitivity of the

pinion profile shifting coefficient parameter is 0.87 % at most 0.1 around the actual

optimum pinion profile shifting coefficient value. When the 0.1 increment of the

pinion profile shifting coefficient is considered sensitivity of the pinion profile shifting

coefficient in a decreasing manner if the pinion profile shifting coefficient values are

moved away from the optimum pinion profile shifting coefficient point.

The optimum center distance and the optimum pinion profile shifting coefficient value

are obtained as 407.678 𝑚𝑚 and 0.2 respectively in the sixth case. It is seen that the

optimum center distances are 407.678 𝑚𝑚 and 431.457 𝑚𝑚 when the pinion profile

shifting coefficient values are adjusted to 0.1 and 0.3, respectively. Therefore,

changing the pinion profile shifting coefficient value 0.1 around the actual optimum

pinion profile shifting coefficient deviates the center distance value 23.78 𝑚𝑚 at

most. The sensitivity of the pinion profile shifting coefficient parameter is 5.83 % at

most 0.1 around the actual optimum pinion profile shifting coefficient value. When

the 0.1 increment of the pinion profile shifting coefficient is considered sensitivity of

the pinion profile shifting coefficient is in a decreasing manner if the pinion profile

shifting coefficient values are moved away from the optimum pinion profile shifting

coefficient point.

92


of external gear pair at relatively higher pinion rotational speed (4000 rpm)


of external gear pair at relatively higher pinion rotational speed (1500 rpm)

-2

0

2

4

6

8

10

12

14

290

300

310

320

330

340

350

-0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)

Profile Shifting Coefficient


-1

0

1

2

3

4

5

6

405

410

415

420

425

430

435

-0,2 -0,1 0 0,1 0,2 0,3 0,4

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)



93

The pressure angle sensitivity and the center distance relation at 4000 rpm and 1500

rpm are given in Figure 4-7 and Figure 4-8, respectively. The optimum center distance

and the optimum pressure angle value are obtained as 298.86 𝑚𝑚 and 23°

respectively in the first case. It is seen that the optimum center distances are

301.472 𝑚𝑚 and 298.86 𝑚𝑚 when the pressure angle values are adjusted to 22.5°

and 23.5°, respectively. Therefore, changing the pressure angle value 0.5° around the

actual optimum pressure angle deviates the center distance value 2.61 𝑚𝑚 at most.

The sensitivity of the pressure angle parameter is 0.87 % at most 0.5° around the

actual optimum pressure angle value. When the 0.5° increment of the pressure angle

is considered sensitivity of the pressure angle in a decreasing manner if the pressure

angle values are moved away from the optimum pressure angle point.

The optimum center distance and the optimum pressure angle value are obtained as


distances are 418.695 𝑚𝑚 and 410.932 𝑚𝑚 when the pressure angle values are

adjusted to 22.5° and 23.5°, respectively. Therefore, changing the pressure angle value

0.5° around the actual optimum pressure angle deviates the center distance value

11.017 𝑚𝑚 at most. The sensitivity of the pressure angle parameter is 2.7 % at most

0.5°around the actual optimum pressure angle value. When the 0.5° increment of the

pressure angle is considered sensitivity of the pressure angle is not always in a

decreasing manner if the pressure angle values are moved away from the optimum

pressure angle point. The sensitivity value does not show a consistent behavior as

opposite to obtained in 4000 𝑟𝑝𝑚. Therefore, when the rotational speed of the pinion

is decreased the pressure angle does not keep its consistent behavior any more.

Although this inconsistency, the sensitivity of the pressure angle is still below the 5 %

0.5°around the actual optimum pressure angle value. The increment of the pressure

angle value is specified as 0.5° which gives a sensitivity value of 2.7 % at most.

94

Figure 4-7 Pressure angle sensitivity and center distance relation of external gear

pair at relatively higher pinion rotational speed (4000 rpm)

Figure 4-8 Pressure angle sensitivity and center distance relation of external gear

pair at relatively lower pinion rotational speed (1500 rpm)

-1

0

1

2

3

4

5

296

298

300

302

304

306

308

310

312

314

316

20 21 22 23 24 25 26

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)

Pressure Angle (degree)


-0,1

0,4

0,9

1,4

1,9

2,4

2,9

406

408

410

412

414

416

418

420

422

21,5 22 22,5 23 23,5 24 24,5 25 25,5 26

Sen

siti

vity

(%

)

Cen

ter

Dis

tan

ce (

mm

)



95

4.1.2. Addendum and Dedendum Radii Sensitivity of External Gear Pairs

Sensitivity analysis of the addendum and dedendum radii is evaluated in this section.

Initial assignments of the design variables and pinion and gear rotational speeds and

input power are given in Table 4.9. In the EX2 case, all the design parameters are

almost kept as the same as they are in the EX1 case except the increment of the

addendum and dedendum radii. In EX3 case, all the design parameters are kept as the

same as they are in the EX1 case except the increment of the addendum and dedendum

radii.


EX1, EX2 and EX3

PARAMATERS EX 1 EX 2 EX 3

𝑁1 10 ∶ 1 ∶ 50 13 ∶ 1 ∶ 47 10 ∶ 1 ∶ 50

𝑚𝑛 (mm) 2 ∶ 0.3 ∶ 7.7 3.2 ∶ 0.3 ∶ 7.7 2.0 ∶ 0.3 ∶ 7.7

𝛼𝑛 (degree) 15 ∶ 1 ∶ 35 15 ∶ 1 ∶ 35 15 ∶ 1 ∶ 35

𝛽 (degree) 5 ∶ 2 ∶ 21 5 ∶ 2 ∶ 21 5 ∶ 2 ∶ 21

𝑥1 −0.6 ∶ 0.2 ∶ 0.6 −0.6 ∶ 0.2 ∶ 0.6 −0.6 ∶ 0.2 ∶ 0.6

𝑦1 0.80 0.80 0.80

𝑦2 1.20 1.20 1.20

𝑦3 0.80 0.80 0.80

𝑦4 1.20 1.20 1.20

𝑦5 −1.50 −1.50 −1.50

𝑦6 −1.10 −1.10 −1.10

𝑦7 −1.50 −1.50 −1.50

𝑦8 −1.10 −1.10 −1.10

𝒊𝒏𝒄𝟏 0.2 0.1 0.05 𝒊𝒏𝒄𝟐 0.2 0.1 0.05 𝒊𝒏𝒄𝟑 0.2 0.1 0.05 𝒊𝒏𝒄𝟒 0.2 0.1 0.05 𝒏𝟏 4000 𝑟𝑝𝑚 4000 𝑟𝑝𝑚 4000 𝑟𝑝𝑚 𝒏𝟐 1017.81 𝑟𝑝𝑚 1017.81 𝑟𝑝𝑚 1017.81 𝑟𝑝𝑚

𝑃𝑜𝑤𝑒𝑟 1700 𝑘𝑊 1700 𝑘𝑊 1700 𝑘𝑊

96

When the optimized results of the Case EX1, Case EX2 and Case EX3 are considered

It is seen that the optimum center distances are 307.07 𝑚𝑚, 298.860 𝑚𝑚 and

296.493 𝑚𝑚, respectively. The optimum center is obtained as 298.860 𝑚𝑚 in the

first case. Therefore, changing the increment value of the addendum and dedendum

radii 0.05 𝑚𝑚 around the actual optimum increment of the addendum and dedendum

radii the center distance value 8.21 𝑚𝑚 at most. The sensitivity of the increment of

the addendum and dedendum radii is 2.75 % at most 0.05 𝑚𝑚 around the actual

optimum increment of the addendum and dedendum radii.

Table 4.10 Optimized design variables (Case EX1 – Case Ex3)

Case

Study

EX 1 EX 2 EX 3

𝒎𝒏 3.2 𝑚𝑚 3.2 𝑚𝑚 3.2 𝑚𝑚

𝜷 13° 13° 21°

𝒙𝟏 0 0.2 0.2

𝒓𝒂𝟏 65.683 𝑚𝑚 64.698 𝑚𝑚 63.926 𝑚𝑚

𝒓𝒂𝟐 247.955 𝑚𝑚 241.058 𝑚𝑚 239.594 𝑚𝑚

𝒓𝒇𝟏 58.130 𝑚𝑚 56.816 𝑚𝑚 55.528 𝑚𝑚

𝒓𝒇𝟐 240.401 𝑚𝑚 233.176 𝑚𝑚 231.367 𝑚𝑚

𝑵𝟏 38 37 35

𝑵𝟐 149 145 138

𝜶𝒏 24° 23° 23°

𝒂𝒘 307.070 𝑚𝑚 298.860 𝑚𝑚 296.493 𝑚𝑚


input power are given in Table 4.11. In the EX4 case, pinion number of teeth is started

from 1 and it is ended at 50 by increasing its value 1. The module is started from 2 𝑚𝑚

and it is ended at 7.7 𝑚𝑚 by increasing its value 0.3 𝑚𝑚. The pressure angle is started

from 15° and it is ended at 35° by increasing its value 1°. The pinion profile shifting

coefficient is started from −0.6 and it is ended at 0.6 by increasing its value 0.2. The

helix angle is started from 5° and it is ended at 21° by increasing its value 2°.

97

In the EX5 case, all the design parameters are kept as the same as they are in the EX1

case except the increment of the addendum and dedendum radii. In EX6 case, all the

design parameters are kept as the same as they are in the EX1 case except the

increment of the addendum and dedendum radii.


EX4, EX5 and EX6

PARAMATERS EX 4 EX 5 EX 6

𝑁1 10 ∶ 1 ∶ 50 10 ∶ 1 ∶ 50 10 ∶ 1 ∶ 50

𝑚𝑛 (mm) 2 ∶ 0.3 ∶ 7.7 2 ∶ 0.3 ∶ 7.7 2.0 ∶ 0.3 ∶ 7.7

𝛼𝑛 (degree) 15 ∶ 1 ∶ 35 15 ∶ 1 ∶ 35 15 ∶ 1 ∶ 35

𝛽 (degree) 5 ∶ 2 ∶ 21 5 ∶ 2 ∶ 21 5 ∶ 2 ∶ 21

𝑥1 −0.6 ∶ 0.2 ∶ 0.6 −0.6 ∶ 0.2 ∶ 0.6 −0.6 ∶ 0.2 ∶ 0.6

𝑦1 0.80 0.80 0.80

𝑦2 1.20 1.20 1.20

𝑦3 0.80 0.80 0.80

𝑦4 1.20 1.20 1.20

𝑦5 −1.50 −1.50 −1.50

𝑦6 −1.10 −1.10 −1.10

𝑦7 −1.50 −1.50 −1.50

𝑦8 −1.10 −1.10 −1.10


𝑃𝑜𝑤𝑒𝑟 1700 𝑘𝑊 1700 𝑘𝑊 1700 𝑘𝑊

The optimized results of the Case EX4, Case EX5 and Case EX6 are given in Table

4.12. It is seen that when the increments of the addendum and dedendum radii are

taken as 0.2 𝑚𝑚, 0.1 𝑚𝑚 𝑎𝑛𝑑 0.05 𝑚𝑚 the optimum center distances are same.

Changing the increment of the addendum and dedendum radii has not an effect on the

center distance when the rotational speed of the pinion is decreased.

98

Table 4.12 Optimized design variables (Case EX4 – Case Ex6)

Case

Study

Inc 0.2 Inc 0.1 Inc 0.05

𝒎𝒏 4.4 𝑚𝑚 4.4 𝑚𝑚 4.4 𝑚𝑚

𝜷 21° 21° 21°

𝒙𝟏 0.2 0.2 0.2

𝒓𝒂𝟏 88.134 𝑚𝑚 88.134 𝑚𝑚 88.134 𝑚𝑚

𝒓𝒂𝟐 329.913 𝑚𝑚 329.913 𝑚𝑚 329.913 𝑚𝑚

𝒓𝒇𝟏 76.351 𝑚𝑚 75.880 𝑚𝑚 75.880 𝑚𝑚

𝒓𝒇𝟐 318.130 𝑚𝑚 318.130 𝑚𝑚 318.130 𝑚𝑚

𝑵𝟏 35 35 35

𝑵𝟐 138 138 138

𝜶𝒏 22° 22° 22°

𝒂𝒘 407.678 𝑚𝑚 407.678 𝑚𝑚 407.678 𝑚𝑚

4.1.3. Detailed Optimization of Case 1

As discussed in the previous sections, the increment of the all design parameters are

specified according to sensitivity analyses. In this section, the specified increments for

each design parameter are used to obtain more optimum results. Initial assignments of

the design variables, rotational speed of the pinion and the gear and the input power

is given in Table 4.13. Optimization result of the first case study is given in

Table 4.4. The first case study is evaluated again by changing the increments of each

design variables. In Case 1 Detailed A, increment of the module, pressure angle, helix

angle and pinion profile shifting coefficient are changed from 0.3 𝑚𝑚 , 1°, 2° and 0.2

to 0.1 𝑚𝑚 , 0.5°, 1° and 0.1 respectively to observe the more optimal results. In Case

1 Detailed B, the increment of the design variables are kept constant as in Case 1

Detailed A. However, increment of the addendum and dedendum radii is changed

from 0.1 𝑚𝑚 to 0.05 𝑚𝑚.

99


Detailed A and Case 1 Detailed B

PARAMATERS Case 1 Case 1 Detailed A

Case 1 Detailed B

𝑁1 10 ∶ 1 ∶ 50 35 ∶ 1 ∶ 41 33 ∶ 1 ∶ 38

𝑚𝑛 (mm) 2 ∶ 0.3 ∶ 7.7 2.9 ∶ 0.1 ∶ 3.4 3.1 ∶ 0.1 ∶ 3.3

𝛼𝑛 (degree) 15 ∶ 1 ∶ 35 21 ∶ 0.5 ∶ 28 22.5 ∶ 0.5 ∶ 26

𝛽 (degree) 5 ∶ 2 ∶ 21 10 ∶ 1 ∶ 20 13 ∶ 1 ∶ 20

𝑥1 −0.6 ∶ 0.2 ∶ 0.6 −0.1 ∶ 0.1 ∶ 0.3 0.1 ∶ 0.1 ∶ 0.2

𝑦1 0.80 0.80 0.80

𝑦2 1.20 1.20 1.20

𝑦3 0.80 0.80 0.80

𝑦4 1.20 1.20 1.20

𝑦5 −1.50 −1.50 −1.50

𝑦6 −1.10 −1.10 −1.10

𝑦7 −1.50 −1.50 −1.50

𝑦8 −1.10 −1.10 −1.10


𝑃𝑜𝑤𝑒𝑟 1700 𝑘𝑊 1700 𝑘𝑊 1700 𝑘𝑊

The optimized results of the Case 1, Case 1 Detailed A and Case 1 Detailed B are

given in Table 4.14. As seen from Table 4.14, the more optimal results are obtained

when the increment of the design variable and the increment of the addendum and

dedendum radii are deceased to values which are obtained as results of sensitivity

analyses.

100


Case

Study

Case 1 Case 1 Detailed A Case 1 Detailed B

𝒎𝒏 3.2 𝑚𝑚 3.3 𝑚𝑚 3.3 𝑚𝑚

𝜷 13° 15° 14°

𝒙𝟏 0.2 0.1 0.1

𝒓𝒂𝟏 64.698 𝑚𝑚 63.204 𝑚𝑚 63.089 𝑚𝑚

𝒓𝒂𝟐 241.058 𝑚𝑚 238.807 𝑚𝑚 237.732 𝑚𝑚

𝒓𝒇𝟏 56.816 𝑚𝑚 55.346 𝑚𝑚 55.437 𝑚𝑚

𝒓𝒇𝟐 233.176 𝑚𝑚 230.949 𝑚𝑚 229.739 𝑚𝑚

𝑵𝟏 37 35 35

𝑵𝟐 145 138 138

𝜶𝒏 23° 25.5° 25°

𝒂𝒘 298.860 𝑚𝑚 295.520 𝑚𝑚 294.189 𝑚𝑚

4.1.4. Detailed Optimization of Case 6

Initial assignments of the design variables, rotational speed of the pinion and the gear

and the input power is given in Table 4.15. Optimization result of the sixth case study

is given in Table 4.8. The sixth case study is evaluated again by changing the

increments of each design variables. In Case 6 Detailed A, increment of the module,

pressure angle, helix angle and pinion profile shifting coefficient are changed from

0.3 𝑚𝑚 , 1°, 2° and 0.2 to 0.1 𝑚𝑚 , 0.5°, 1° and 0.1 respectively to observe the more

optimal results. In Case 6 Detailed B, the increment of the design variables are kept

constant. However, increment of the addendum and dedendum radii are changed from

0.1 𝑚𝑚 to 0.05 𝑚𝑚.

101


Detailed A and Case 6 Detailed B


Case 6 Detailed B

𝑁1 10 ∶ 1 ∶ 50 27 ∶ 1 ∶ 41 33 ∶ 1 ∶ 38

𝑚𝑛 (mm) 2 ∶ 0.3 ∶ 7.7 4.0 ∶ 0.1 ∶ 5.6 3.1 ∶ 0.1 ∶ 3.3

𝛼𝑛 (degree) 15 ∶ 1 ∶ 35 20 ∶ 0.5 ∶ 28 22.5 ∶ 0.5 ∶ 26

𝛽 (degree) 5 ∶ 2 ∶ 21 7 ∶ 1 ∶ 21 13 ∶ 1 ∶ 20

𝑥1 −0.6 ∶ 0.2 ∶ 0.6 −0.1 ∶ 0.1 ∶ 0.3 0.1 ∶ 0.1 ∶ 0.2

𝑦1 0.80 0.80 0.80

𝑦2 1.20 1.20 1.20

𝑦3 0.80 0.80 0.80

𝑦4 1.20 1.20 1.20

𝑦5 −1.50 −1.50 −1.50

𝑦6 −1.10 −1.10 −1.10

𝑦7 −1.50 −1.50 −1.50

𝑦8 −1.10 −1.10 −1.10


𝑃𝑜𝑤𝑒𝑟 1700 𝑘𝑊 1700 𝑘𝑊 1700 𝑘𝑊

The optimized results of the Case 1, Case 1 Detailed A and Case 1 Detailed B are

given in Table 4.16. As seen from Table 4.16, the more optimal results are obtained

when the increment of the design variable and the increment of the addendum and

dedendum radii are deceased to values which are obtained as results of sensitivity

analyses.

102


Case

Study

Case 6 Actual Case 6 A

Detailed

Actual Case 6 B

Detailed

𝒎𝒏 4.4 𝑚𝑚 4.2 𝑚𝑚 4.4 𝑚𝑚

𝜷 21° 20° 20°

𝒙𝟏 0.2 0.2 0.2

𝒓𝒂𝟏 88.134 𝑚𝑚 88.050 𝑚𝑚 87.561 𝑚𝑚

𝒓𝒂𝟐 329.913 𝑚𝑚 328.512 𝑚𝑚 327.533 𝑚𝑚

𝒓𝒇𝟏 75.880 𝑚𝑚 76.876 𝑚𝑚 75.620 𝑚𝑚

𝒓𝒇𝟐 318.130 𝑚𝑚 317.338 𝑚𝑚 316.061 𝑚𝑚

𝑵𝟏 35 37 35

𝑵𝟐 138 145 138

𝜶𝒏 22° 22° 23°

𝒂𝒘 407.678 𝑚𝑚 406.729 𝑚𝑚 405.026 𝑚𝑚

4.2. Sensitivity Analysis and Case Studies of Internal Gear Pair

The internal gear pairs are examined in the case studies from 11 to 20. The rotational

speed of the pinion is taken as 900 𝑟𝑝𝑚 in the cases from 11 to 15. The pinion

rotational speed is taken as 3000 𝑟𝑝𝑚 in the cases from 16 to 20. Difference between

the pinion rotational speed is applied to observe the relatively high and low speeds on

the sensitivity of the design variables. Table 4.17 gives the pinion and gear rotational

speeds and the power input in the cases from 11 to 15. Table 4.18 gives the upper and

lower bounds and the increment values of the pinion and gear addendum and

dedendum radii.


Parameter Value

𝑛1 900 𝑟𝑝𝑚

𝑛2 329 𝑟𝑝𝑚


103




0.80 0.1 1.20


−1.20 0.1 −0.80


−1.40 0.1 −1.00


1.00 0.1 1.40

Initial assignments of the design variables are given in Table 4.19. In the 11th case,






by increasing its value 2°. In the 12th cases, all the design parameters are kept as the

same as they are in the 11th case except the parameter of the module. The module


module on objective function. In the 13th cases, all the design parameters are kept as

the same as they are in the 11th case except the parameter of the helix angle. The helix


of the helix angle on objective function. In the 14th cases, all the design parameters are

kept as the same as they are in the 11th case except the parameter of the pinion profile

shifting coefficient. The pinion profile shifting coefficient values are taken as constant

values to observe the effect of the increment value of the pinion profile shifting

coefficient on objective function. In the 15th cases, all the design parameters are kept

as the same as they are in the 11th case except the parameter of the pressure angle. The

pressure angle values are taken as constant values to observe the effect of the

increment value of the pressure angle on objective function.

104

Table 4.19 Input parameters of the studies from Case 11 to Case 15

Case 𝑵𝟏 𝒎𝒏 (𝒎𝒎) 𝜶𝒏(𝒅𝒆𝒈) 𝜷(𝒅𝒆𝒈) 𝒙𝟏

11 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

12 A 10 ∶ 1 ∶ 50 6.8 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

12 B 10 ∶ 1 ∶ 50 6.9 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

12 C 10 ∶ 1 ∶ 50 7.0 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

12 D 10 ∶ 1 ∶ 50 7.1 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

12 E 10 ∶ 1 ∶ 50 7.2 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

12 F 10 ∶ 1 ∶ 50 7.3 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

12 G 10 ∶ 1 ∶ 50 7.4 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

12 H 10 ∶ 1 ∶ 50 7.5 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

13 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 6 −0.6 ∶ 0.2 ∶ 0.6

13 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 7 −0.6 ∶ 0.2 ∶ 0.6

13 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 8 −0.6 ∶ 0.2 ∶ 0.6

13 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 9 −0.6 ∶ 0.2 ∶ 0.6

13 E 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 10 −0.6 ∶ 0.2 ∶ 0.6

13 F 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 11 −0.6 ∶ 0.2 ∶ 0.6

13 G 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 12 −0.6 ∶ 0.2 ∶ 0.6

13 H 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 13 −0.6 ∶ 0.2 ∶ 0.6

13 I 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 14 −0.6 ∶ 0.2 ∶ 0.6

13 J 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 15 −0.6 ∶ 0.2 ∶ 0.6

13 K 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 16 −0.6 ∶ 0.2 ∶ 0.6

13 L 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 17 −0.6 ∶ 0.2 ∶ 0.6

14 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.1

14 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0

14 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0.1

14 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0.2

15 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 27.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 28 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 28.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 29 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 E 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 29.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 F 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 30 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 G 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 30.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 H 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 31 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 I 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 31.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 J 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 32 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 K 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 32.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

15 L 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 33 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

105

Table 4.20 Optimized design variables (Case 11- Case15 )


𝜶𝒏

𝜷


𝒓𝒂𝟐 𝒓𝒇𝟏 𝒓𝒇𝟐

𝒂𝒘


11 25 68 7.1 29 7 0 95.855 236.060 80.117 251.797 153.796

12 A 26 71 6.8 28 7 0.2 96.600 237.732 81.528 252.804 154.149

12 B 25 68 6.9 28 15 0 95.722 235.732 80.006 251.448 153.583

12 C 25 68 7.0 28 13 0 96.267 237.076 80.462 252.881 154.459

12 D 25 68 7.1 29 7 0 95.855 236.060 80.117 251.797 153.796

12 E 25 68 7.2 29 5 0 96.849 238.508 80.948 254.408 155.391

12 F 25 68 7.3 29 5 0 98.194 241.820 82.072 257.942 157.550

12 G 23 63 7.4 26 19 0.2 98.613 239.488 80.612 257.488 156.528

12 H 23 63 7.5 28 15 0 96.281 236.820 79.199 253.902 155.291

13 A 25 68 7.1 29 6 0 95.664 235.591 79.958 251.297 153.491

13 B 25 68 7.1 29 7 0 95.855 236.060 80.117 251.797 153.796

13 C 25 68 7.1 28 8 0.2 97.509 238.037 81.018 253.810 154.150

13 D 25 68 7.1 29 9 0 96.326 237.221 80.511 253.035 154.553

13 E 25 68 7.1 29 10 0 96.608 237.915 80.747 253.775 155.005

13 F 35 96 5.0 27 11 0.2 94.741 240.417 83.025 251.623 155.354

13 G 35 96 5.0 27 12 0.2 95.078 241.272 83.321 252.518 155.907

13 H 31 85 5.6 28 13 0 94.256 238.513 81.612 251.157 155.177

13 I 31 85 5.6 28 14 0 94.652 239.515 81.954 252.212 155.829

13 J 37 101 4.7 27 15 0.2 95.370 241.830 84.178 252.535 155.706

13 K 37 101 4.7 27 16 0 94.855 242.026 84.098 252.782 156.461

13 L 29 79 5.9 27 17 0.2 96.245 238.763 82.672 252.336 154.240

14 A 37 101 4.7 29 15 -0.1 93.910 240.857 83.692 251.075 155.706

14 B 25 68 7.1 29 7 0 95.854 236.060 80.117 251.797 153.796

14 C 29 79 5.9 27 17 0.1 95.629 80.117 82.055 251.719 154.240

14 D 26 71 6.8 28 7 0.2 96.600 251.797 81.528 252.804 154.149

15 A 29 79 5.9 27.5 17 0 95.012 237.529 81.438 251.102 154.240

15 B 26 71 6.8 28 7 0.2 96.600 237.732 81.528 252.804 154.149

15 C 25 68 7.1 28.5 7 0 95.854 236.060 80.117 251.797 153.796

15 D 25 68 7.1 29 7 0 95.854 236.060 80.117 251.797 153.796

15 E 50 137 4.4 29.5 5 0 114.395 298.576 105.120 307.852 192.131

15 F 50 137 3.8 30 21 0 105.015 275.563 96.874 283.296 177.060

15 G 40 109 4.4 30.5 21 0 98.031 253.090 88.605 262.045 162.600

15 H 34 93 5 31 21 0 95.332 244.757 84.620 254.933 157.994

15 I 37 101 4.7 31.5 17 0 94.855 244.263 85.025 253.601 157.272

15 J 35 96 5 32 13 0 93.907 242.208 84.157 251.958 156.511

15 K 34 93 5.3 32.5 9 0 95.516 245.229 84.784 255.425 158.300

15 L 34 93 5.9 33 5 0 105.421 270.660 94.168 281.913 174.715

106

The optimized results of the cases from 11 to 15 are given in Table 4.20. Table 4.21

gives the pinion and gear rotational speeds and the power input in the cases from 16

to 20. Table 4.22 gives the upper and lower bounds and the increment values of the

pinion and gear addendum and dedendum radii.


Parameter Value

𝑛1 3000 𝑟𝑝𝑚

𝑛2 1096.67 𝑟𝑝𝑚





0.80 0.1 1.20


−1.20 0.1 −0.80


−1.40 0.1 −1.00


1.00 0.1 1.40

Initial assignments of the design variables are given in Table 4.23. In the 16th case,





0.6 by increasing its value 0.2. The helix angle is started from 5° and it is ended at

21°by increasing its value 2°. In the 17th cases, all the design parameters are kept as

the same as they are in the 16th case except the parameter of the module. The module


module on objective function. In the 18th, all the design parameters are kept as the

same as they are in the 16th case except the parameter of the helix angle. The helix

107


of the helix angle on objective function. In the 19th cases, all the design parameters are

kept as the same as they are in the 16th case except the parameter of the pinion profile

shifting coefficient. The pinion profile shifting coefficient values are taken as constant

values to observe the effect of the increment value of the pinion profile shifting

coefficient on objective function. In the 20th cases, all the design parameters are kept

as the same as they are in the 16th case except the parameter of the pressure angle. The

pressure angle values are taken as constant values to observe the effect of the

increment value of the pressure angle on objective function.

108

Table 4.23 Input parameters of the studies from (Case 16 – Case 20)

Case 𝑵𝟏 𝒎𝒏 (𝒎𝒎) 𝜶𝒏(𝒅𝒆𝒈) 𝜷(𝒅𝒆𝒈) 𝒙𝟏

16 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

17 A 10 ∶ 1 ∶ 50 4.3 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

17 B 10 ∶ 1 ∶ 50 4.4 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

17 C 10 ∶ 1 ∶ 50 4.5 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

17 D 10 ∶ 1 ∶ 50 4.6 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

17 E 10 ∶ 1 ∶ 50 4.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

17 F 10 ∶ 1 ∶ 50 4.8 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

17 G 10 ∶ 1 ∶ 50 4.9 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

17 H 10 ∶ 1 ∶ 50 5.0 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

18 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 8 −0.6 ∶ 0.2 ∶ 0.6

18 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 9 −0.6 ∶ 0.2 ∶ 0.6

18 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 10 −0.6 ∶ 0.2 ∶ 0.6

18 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 11 −0.6 ∶ 0.2 ∶ 0.6

18 E 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 12 −0.6 ∶ 0.2 ∶ 0.6

18 F 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 13 −0.6 ∶ 0.2 ∶ 0.6

18 G 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 14 −0.6 ∶ 0.2 ∶ 0.6

18 H 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 15 −0.6 ∶ 0.2 ∶ 0.6

19 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 −0.1

19 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0

19 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0.1

19 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 15 ∶ 1 ∶ 35 5 ∶ 2 ∶ 21 0.2

20 A 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 24.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

20 B 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 25 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

20 C 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 25.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

20 D 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 26 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

20 E 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 26.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

20 F 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 27 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

20 G 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 27.5 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

20 H 10 ∶ 1 ∶ 50 2 ∶ 0.3 ∶ 7.7 28 5 ∶ 2 ∶ 21 −0.6 ∶ 0.2 ∶ 0.6

109

Table 4.24 Optimized design variables (Case 16 – Case 20)


𝜶𝒏

𝜷


𝒓𝒂𝟐

𝒓𝒇𝟏

𝒓𝒇𝟐

𝒂𝒘


16 25 68 4.7 26 11 0.2 65.116 158.482 53.625 169.494 102.941

17 A 26 71 4.3 24 21 0.2 64.944 159.365 54.350 169.959 103.633

17 B 26 71 4.4 25 17 0.2 64.875 159.196 53.832 169.779 103.524

17 C 26 71 4.5 26 11 0.2 64.638 158.614 53.635 169.158 103.145

17 D 25 68 4.6 25 17 0.2 65.418 159.217 54.355 170.280 103.419

17 E 25 68 4.7 26 11 0.2 65.116 158.482 54.104 169.494 102.941

17 F 25 68 4.8 26 7 0.2 65.770 160.073 54.647 171.196 103.975

17 G 23 63 4.9 25 19 0.2 65.298 158.580 52.860 170.499 103.647

17 H 23 63 5.0 25 17 0.2 65.879 159.991 53.330 172.016 104.569

18 A 37 101 3.2 25 8 0 63.013 159.634 55.258 167.389 103.406

18 B 37 101 3.2 25 9 0 63.178 160.051 55.402 167.826 103.676

18 C 25 68 4.7 26 10 0.2 64.906 157.970 53.452 168.947 102.609

18 D 25 68 4.7 26 11 0.2 65.116 158.482 53.625 169.494 102.941

18 E 25 68 4.7 26 12 0.2 65.348 159.050 53.816 170.097 103.308

18 F 25 68 4.7 26 13 0.2 65.601 159.662 54.507 170.757 103.708

18 G 25 68 4.7 26 14 0.2 65.877 160.333 54.252 171.474 104.144

18 H 25 68 4.7 26 15 0.2 66.175 161.058 54.497 172.249 104.615

19 A 25 68 4.7 25 11 -0.1 63.680 157.524 53.146 168.058 102.941

19 B 25 68 4.7 25 11 0 64.159 158.003 53.146 168.537 102.941

19 C 25 68 4.7 25 11 0.1 64.159 158.961 54.583 168.537 102.941

19 D 25 68 4.7 25 11 0.2 65.116 158.482 53.625 169.494 102.941

20 A 37 101 3.2 24.5 9 0 63.178 160.051 55.402 167.826 103.676

20 B 26 71 4.4 25 17 0.2 64.875 159.196 53.832 169.779 103.524

20 C 25 68 4.7 25.5 11 0.2 65.116 158.482 53.625 169.494 102.941

20 D 25 68 4.7 26 11 0.2 65.116 158.482 53.625 169.494 102.941

20 E 26 71 4.4 26.5 11 0 63.955 158.736 53.372 168.858 103.524

20 F 25 68 4.7 27 11 0.2 65.116 158.961 54.583 169.494 102.941

20 G 25 68 4.7 27.5 11 0 64.159 158.003 53.146 168.537 102.941

20 H 25 68 4.7 28 11 0 64.159 158.003 53.146 168.537 102.941

110

The optimized results of the cases from 16 to 20 are given in Table 4.24. The module

sensitivity and the center distance relation at 900 rpm and 3000 rpm are given in Figure

4-9 and Figure 4-10, respectively. The optimum center distance and the optimum

module value are obtained as 153.796 𝑚𝑚 and 7.1 mm respectively in the 11th case.

It is seen that the optimum center distances are 154.459 𝑚𝑚 and 155.391 𝑚𝑚 when

the module values are adjusted to 7.0 𝑚𝑚 and 7.2 𝑚𝑚, respectively. Therefore,

changing the module value 0.1 𝑚𝑚 around the actual optimum module deviates the

center distance value 1.59 𝑚𝑚 at most. The sensitivity of the module parameter is

1.04 % at most 0.1 𝑚𝑚 around the actual optimum module value. When the 0.1 𝑚𝑚

increment of the module is considered sensitivity of the module is not always in a

decreasing manner if the module values are moved away from the optimum module

point. When the module is moved away from the actual optimum point there can be

local optimum points as observed at 7.50 𝑚𝑚 module.

The optimum center distance and the optimum module value are obtained as

102.941 𝑚𝑚 and 4.7 mm respectively in the 16th case. It is seen that the optimum

center distances are 103.419 𝑚𝑚 and 103.975 𝑚𝑚 when the module values are

adjusted to 4.6 𝑚𝑚 and 4.7 𝑚𝑚, respectively. Therefore, changing the module value

0.1 𝑚𝑚 around the actual optimum module deviates the center distance value

1.034 𝑚𝑚 at most. The sensitivity of the module parameter is 1.00 % at most

0.1 𝑚𝑚 around the actual optimum module value. When the 0.1 𝑚𝑚 increment of the

module is considered sensitivity of the module is in a decreasing manner if the module

values are moved away from the optimum module point. The increment of the module

value is specified as 0.1 𝑚𝑚 which gives a sensitivity value of 1.00 % at most.

111


relatively lower pinion rotational speed (900 rpm)


relatively higher pinion rotational speed (3000 rpm)

6,75 6,85 6,95 7,05 7,15 7,25 7,35 7,45 7,55

-0,5

0

0,5

1

1,5

2

2,5

3

153

153,4

153,8

154,2

154,6

155

155,4

155,8

156,2

156,6

157

157,4

157,8

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)

module (mm)


4,25 4,35 4,45 4,55 4,65 4,75 4,85 4,95 5,05

-0,1

0,1

0,3

0,5

0,7

0,9

1,1

1,3

1,5

1,7

102,5

103

103,5

104

104,5

105

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)

module (mm)


112

The helix angle sensitivity and the center distance relation at 900 rpm and 3000 rpm

are given in Figure 4-11 and Figure 4-12 respectively. The optimum center distance

and the optimum helix angle value are obtained as 153.796 𝑚𝑚 and 7° respectively

in the first case. It is seen that the optimum center distances are 153.491 𝑚𝑚 and

154.15 𝑚𝑚 when the helix angle values are adjusted to 6° and 8°, respectively.

Therefore, changing the helix angle value 1° around the actual optimum helix angle

deviates the center distance value 0.36 𝑚𝑚 at most. The sensitivity of the helix angle

parameter is 0.23 % at most 1° around the actual optimum helix angle value. When

the 1° increment of the helix angle is considered sensitivity of the helix angle is not in

a decreasing manner if the helix angle values are moved away from the optimum helix

angle point. The sensitivity of the helix angle has an inconsistent behavior at relatively

lower speed like it is also observed in external gear pairs as seen in Figure 4-4.

The optimum center distance and the optimum helix angle value are obtained as


distances are 102.609 𝑚𝑚 and 103.308 𝑚𝑚 when the helix angle values are adjusted

to 10° and 12°, respectively. Therefore, changing the helix angle value 1° around the

actual optimum helix angle deviates the center distance value 0.37 𝑚𝑚 at most. The

sensitivity of the helix angle parameter is 0.35 % at most 1°around the actual

optimum helix angle value. When the 1° increment of the helix angle is considered

sensitivity of the module is not always in a decreasing manner if the helix angle values

are moved away from the optimum helix angle point. When the 1° increment of the

helix angle is considered sensitivity of the helix angle is in a decreasing manner if the

pinion helix angle values are moved away from the optimum pinion profile helix angle

point. The increment of the helix angle value is specified as 1° which gives a

sensitivity value of 0.35 % at most.

113


at relatively lower pinion rotational speed (900 rpm)


at relatively higher pinion rotational speed (3000 rpm)

6 8 10 12 14 16 18

-0,4

0,1

0,6

1,1

1,6

2,1

153,1

153,6

154,1

154,6

155,1

155,6

156,1

156,6

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)



6 8 10 12 14 16

-0,5

0

0,5

1

1,5

2

102,3

102,7

103,1

103,5

103,9

104,3

104,7

Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)



114

The pinion profile shifting coefficient sensitivity and the center distance relation at

900 is given in Figure 4-13. The optimum center distance and the optimum profile

shifting coefficient are obtained as 153.796 𝑚𝑚 and 0 respectively in the 11th case.

It is seen that the optimum center distances are 155.706 𝑚𝑚 and 154.24 𝑚𝑚 when

the pinion profile shifting coefficient values are adjusted to −0.1 and 0.1, respectively.

Therefore, changing the pinion profile shifting coefficient value 0.1 around the actual

optimum profile shifting coefficient deviates the center distance value 1.91 𝑚𝑚 at

most. The sensitivity of the pinion profile shifting coefficient parameter is 1.24 % at

most 0.1 around the actual optimum pinion profile shifting coefficient value. When

the 0.1 increment of the pinion profile shifting coefficient is considered sensitivity of

the pinion profile shifting coefficient in a decreasing manner if the pinion profile

shifting coefficient values are moved away from the optimum pinion profile shifting

coefficient point. When Table 4.24 is considered, it is seen that changing of the pinion

profile shifting coefficient does not effect the value of the optimum center distance in

internal gear pairs at relatively high speed. The increment of the pinion profile shifting

coefficient value is specified as 0.1 which gives a sensitivity value of 1.24 % at most.

Figure 4-13 Pinion profile shifting coefficient sensitivity and center distance

relation of internal gear pair at relatively lower pinion rotational speed (900 rpm)

-0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2 0,25-0,2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

153,4

153,8

154,2

154,6

155

155,4

155,8


Sen

siti

vity

(%

)

Ce

nte

r D

ista

nce

(m

m)


115

The pressure angle sensitivity and the center distance relation at 900 rpm and 3000

rpm are given in Figure 4-14 and Figure 4-15, respectively. The optimum center

distance and the optimum pressure angle value are obtained as 153.796 𝑚𝑚 and 29°

respectively in the 11th case. It is seen that the optimum center distances are

153.796 𝑚𝑚 and 192.131 𝑚𝑚 when the pressure angle values are adjusted to 28.5°

and 29.5°, respectively. Therefore, changing the pressure angle value 0.5° around the

actual optimum pressure angle deviates the center distance value 38.33 𝑚𝑚 at most.

The sensitivity of the pressure angle parameter is 24.93 % at most 0.5° around the

actual optimum pressure angle value. When the 0.5° increment of the pressure angle

is considered sensitivity of the pressure angle is not in a decreasing manner if the

pressure angle values are moved away from the optimum pressure angle point. It is

seen that the sensitivity has sudden jumps in some points. However, when the 28.50°

is considered, the sensitivity of the pressure angle becomes very high. The 0.5°

increment of the pressure angle is enough to obtain the optimum point if the initially

assigned interval is wide enough. The optimum center distance and the optimum

pressure angle value are obtained as 102.941 𝑚𝑚 and 26° respectively in the 16th

case. It is seen that the optimum center distances are 102.941 𝑚𝑚 and 103.524 𝑚𝑚

when the pressure angle values are adjusted to 25.5° and 24.5°, respectively.

Therefore, changing the pressure angle value 0.5° around the actual optimum pressure

angle deviates the center distance value 0.583 𝑚𝑚 at most. The sensitivity of the

pressure angle parameter is 0.57 % at most 0.5°around the actual optimum pressure

angle value. When the 0.5° increment of the pressure angle is considered sensitivity

of the pressure angle is always in a decreasing manner if the pressure angle values are

moved away from the optimum pressure angle point. However, the sensitivity value

does not show a consistent behavior as it is observed in 900 rpm. Therefore, when the

rotational speed of the pinion is decreased the pressure angle does not keep its

consistent behavior any more. Although this inconsistency, the sensitivity of the

pressure angle is still below the 5 % 0.5°around the actual optimum pressure angle

value. The increment of the pressure angle value is specified as 0.5°.

116

Figure 4-14 Pressure angle sensitivity and center distance relation of internal gear

pair at relatively lower pinion rotational speed (900 rpm)

Figure 4-15 Pressure angle sensitivity and center distance relation of internal gear

pair at relatively higher pinion rotational speed (3000 rpm)

27 28 29 30 31 32 33 34

-5

0

5

10

15

20

25

0

50

100

150

200

250

Sen

siti

vity

(%

)

Ce

nte

r D

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(m

m)



24 24,5 25 25,5 26 26,5 27 27,5 28 28,5

-0,2

-0,1

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

102,8

102,9

103

103,1

103,2

103,3

103,4

103,5

103,6

103,7

103,8

Sen

siti

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(%

)

Ce

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(m

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117

4.2.1. Addendum and Dedendum Radii Sensitivity of Internal Gear Pairs

Sensitivity analysis of the addendum and dedendum radii is evaluated in this section.


input power are given in Table 4.25. In the INT1 case, pinion number of teeth is started





helix angle is started from 5° and it is ended at 21° by increasing its value 2°. In the

INT2 case, all the design parameters are kept as the same as they are in the INT1 case

except the increment of the addendum and dedendum radii. In INT3 case, all the

design parameters are kept as the same as they are in the INT1 case except the



INT1, INT2 and INT3

PARAMATERS INT 1 INT 2 INT 3

𝑁1 10 ∶ 1 ∶ 50 10 ∶ 1 ∶ 50 20 ∶ 1 ∶ 40

𝑚𝑛 (mm) 2 ∶ 0.3 ∶ 7.7 2 ∶ 0.3 ∶ 7.7 4.4 ∶ 0.3 ∶ 7.7

𝛼𝑛 (degree) 15 ∶ 1 ∶ 35 15 ∶ 1 ∶ 35 25 ∶ 1 ∶ 35

𝛽 (degree) 5 ∶ 2 ∶ 21 5 ∶ 2 ∶ 21 5 ∶ 2 ∶ 21

𝑥1 −0.6 ∶ 0.2 ∶ 0.6 −0.6 ∶ 0.2 ∶ 0.6 −0.4 ∶ 0.2 ∶ 0.4

𝑦1 0.80 0.80 0.80

𝑦2 1.20 1.20 1.20

𝑦3 −1.20 −1.20 −1.20

𝑦4 −0.80 −0.80 −0.80

𝑦5 −1.40 −1.40 −1.40

𝑦6 −1.00 −1.00 −1.00

𝑦7 1.00 1.00 1.00

𝑦8 1.40 1.40 1.40

𝒊𝒏𝒄𝟏 0.2 0.1 0.05 𝒊𝒏𝒄𝟐 0.2 0.1 0.05 𝒊𝒏𝒄𝟑 0.2 0.1 0.05 𝒊𝒏𝒄𝟒 0.2 0.1 0.05 𝒏𝟏 900 𝑟𝑝𝑚 900 𝑟𝑝𝑚 900 𝑟𝑝𝑚 𝒏𝟐 329 𝑟𝑝𝑚 329 𝑟𝑝𝑚 329 𝑟𝑝𝑚

𝑃𝑜𝑤𝑒𝑟 1700 𝑘𝑊 1700 𝑘𝑊 1700 𝑘𝑊

118

The optimized results of the Case INT1, Case INT2 and Case INT3 are given in Table


taken as 0.2 𝑚𝑚, 0.1 𝑚𝑚 𝑎𝑛𝑑 0.05 𝑚𝑚 the optimum center distances are

157.880 𝑚𝑚, 153.796 𝑚𝑚 and 153.233 𝑚𝑚, respectively. The optimum center is

obtained as 153.796 𝑚𝑚 in the 11th case. Therefore, changing the increment value of

the addendum and dedendum radii 0.05 𝑚𝑚 around the actual optimum increment of

the addendum and dedendum radii the center distance value deviates 4.084 𝑚𝑚 at

most. The sensitivity of the increment of the addendum and dedendum radii is 2.66 %

at most 0.05 𝑚𝑚 around the actual optimum increment of the addendum and

dedendum radii.

Table 4.26 Optimized design variables (Case INT1 – CASE INT3)

Case

Study

INT 1 INT 2 INT 3

𝒎𝒏 5 𝑚𝑚 7.1 𝑚𝑚 7.1 𝑚𝑚

𝜷 15° 7° 5°

𝒙𝟏 0 0 0.2

𝒓𝒂𝟏 94.728 𝑚𝑚 95.854 𝑚𝑚 97.285 𝑚𝑚

𝒓𝒂𝟐 244.325 𝑚𝑚 236.060 𝑚𝑚 235.551 𝑚𝑚

𝒓𝒇𝟏 84.375 𝑚𝑚 80.117 𝑚𝑚 80.180 𝑚𝑚

𝒓𝒇𝟐 254.678 𝑚𝑚 251.797 𝑚𝑚 252.300 𝑚𝑚

𝑵𝟏 35 25 25

𝑵𝟐 96 68 68

𝜶𝒏 32° 29° 26°

𝒂𝒘 157.880 𝑚𝑚 153.796 𝑚𝑚 153.233 𝑚𝑚


input power are given in Table 4.27. In the INT4 case, pinion number of teeth is started




119


helix angle is started from 5° and it is ended at 21° by increasing its value 2°.

In the INT5 case, all the design parameters are kept as the same as they are in the INT4

case except the increment of the addendum and dedendum radii. In INT6 case, all the

design parameters are kept as the same as they are in the INT4 case except the



INT4, INT5 and INT6

PARAMATERS INT 4 INT 5 INT 6

𝑁1 10 ∶ 1 ∶ 50 10 ∶ 1 ∶ 50 20 ∶ 1 ∶ 40

𝑚𝑛 (mm) 2 ∶ 0.3 ∶ 7.7 2 ∶ 0.3 ∶ 7.7 4.1 ∶ 0.3 ∶ 7.7

𝛼𝑛 (degree) 15 ∶ 1 ∶ 35 15 ∶ 1 ∶ 35 25 ∶ 1 ∶ 35

𝛽 (degree) 5 ∶ 2 ∶ 21 5 ∶ 2 ∶ 21 5 ∶ 2 ∶ 21

𝑥1 −0.6 ∶ 0.2 ∶ 0.6 −0.6 ∶ 0.2 ∶ 0.6 −0.4 ∶ 0.2 ∶ 0.4

𝑦1 0.80 0.80 0.80

𝑦2 1.20 1.20 1.20

𝑦3 −1.20 −1.20 −1.20

𝑦4 −0.80 −0.80 −0.80

𝑦5 −1.40 −1.40 −1.40

𝑦6 −1.00 −1.00 −1.00

𝑦7 1.00 1.00 1.00

𝑦8 1.40 1.40 1.40


𝑃𝑜𝑤𝑒𝑟 1700 𝑘𝑊 1700 𝑘𝑊 1700 𝑘𝑊

The optimized results of the Case INT4, Case INT5 and Case INT6 are given in Table


taken as 0.2 𝑚𝑚, 0.1 𝑚𝑚 𝑎𝑛𝑑 0.05 𝑚𝑚 the optimum center distances are

120

105.299 𝑚𝑚, 104.569 𝑚𝑚 and 102.941 𝑚𝑚 respectively. The optimum center is

obtained as 102.941 𝑚𝑚 in the 16th case. Therefore, changing the increment value of

the addendum and dedendum radii 0.05 𝑚𝑚 around the actual optimum increment of

the addendum and dedendum radii the center distance value deviates 1.628 𝑚𝑚 at

most. The sensitivity of the increment of the addendum and dedendum radii is 1.60 %

at most 0.05 𝑚𝑚 around the actual optimum increment of the addendum and

dedendum radii.

Table 4.28 Optimized design variables (Case INT4 – CASE INT6)

Case

Study

INT 4 INT 5 INT 6

𝒎𝒏 3.8 𝑚𝑚 5.0 𝑚𝑚 4.7 𝑚𝑚

𝜷 13° 17° 11°

𝒙𝟏 0 0.2 0.2

𝒓𝒂𝟏 63.569 𝑚𝑚 65.879 𝑚𝑚 65.356 𝑚𝑚

𝒓𝒂𝟐 162.628 𝑚𝑚 159.991 𝑚𝑚 158.242 𝑚𝑚

𝒓𝒇𝟏 55.769 𝑚𝑚 53.330 𝑚𝑚 53.386 𝑚𝑚

𝒓𝒇𝟐 170.428 𝑚𝑚 172.016 𝑚𝑚 169.494 𝑚𝑚

𝑵𝟏 31 23 25

𝑵𝟐 85 63 68

𝜶𝒏 31° 25° 25°

𝒂𝒘 105.299 𝑚𝑚 104.569 𝑚𝑚 102.941 𝑚𝑚

4.2.2. Detailed optimization of Case 11

As discussed in the previous sections, the increment of the all design parameters are

specified according to sensitivity analyses. In this section, the specified increments for

each design parameter are used to obtain more optimum results.

Initial assignments of the design variables, rotational speed of the pinion and the gear

and the input power is given in Table 4.29. Optimization result of the first case study

is given in Table 4.20. The first case study is evaluated again by changing the

increments of each design variables. In Case 11 Detailed A, increment of the module,

121

pressure angle, helix angle and pinion profile shifting coefficient are changed from

0.3 𝑚𝑚 , 1°, 2° and 0.2 to 0.1 𝑚𝑚 , 0.5°, 1° and 0.1 respectively to observe the more

optimal results. In Case 11 Detailed B, the increment of the design variables are kept

constant as in Case 11 Detailed A. However, increment of the addendum and

dedendum radii is changed from 0.1 𝑚𝑚 to 0.05 𝑚𝑚.


11 Detailed A and Case 11 Detailed B


Case 11 Detailed B

𝑁1 10 ∶ 1 ∶ 50 20 ∶ 1 ∶ 40 23 ∶ 1 ∶ 27

𝑚𝑛 (mm) 2 ∶ 0.3 ∶ 7.7 4.0 ∶ 0.1 ∶ 8.0 6.5 ∶ 0.1 ∶ 7.5

𝛼𝑛 (degree) 15 ∶ 1 ∶ 35 25 ∶ 0.5 ∶ 33 25 ∶ 0.5 ∶ 32

𝛽 (degree) 5 ∶ 2 ∶ 21 5 ∶ 1 ∶ 21 5 ∶ 1 ∶ 21

𝑥1 −0.6 ∶ 0.2 ∶ 0.6 −0.2 ∶ 0.1 ∶ 0.4 −0.2 ∶ 0.1 ∶ 0.3

𝑦1 0.80 0.80 0.80

𝑦2 1.20 1.20 1.20

𝑦3 −1.20 −1.20 −1.20

𝑦4 0.8 0.8 0.8

𝑦5 −1.40 −1.40 −1.40

𝑦6 −1.00 −1.00 −1.00

𝑦7 1 1 1

𝑦8 1.4 1.4 1.4

𝒊𝒏𝒄𝟏 0.1 0.1 0.05 𝒊𝒏𝒄𝟐 0.1 0.1 0.05 𝒊𝒏𝒄𝟑 0.1 0.1 0.05 𝒊𝒏𝒄𝟒 0.1 0.1 0.05 𝒏𝟏 900 𝑟𝑝𝑚 900 𝑟𝑝𝑚 900 𝑟𝑝𝑚 𝒏𝟐 329 𝑟𝑝𝑚 329 𝑟𝑝𝑚 329 𝑟𝑝𝑚

𝑃𝑜𝑤𝑒𝑟 1700 𝑘𝑊 1700 𝑘𝑊 1700 𝑘𝑊

122


Case

Study

Case 11 Case 11 Detailed

A

Case 11 Detailed

B

𝒎𝒏 7.1 𝑚𝑚 7.1 𝑚𝑚 7.1 𝑚𝑚

𝜷 7° 28.5° 5°

𝒙𝟏 0 0.1 0.2

𝒓𝒂𝟏 95.855 𝑚𝑚 96.216 𝑚𝑚 97.285 𝑚𝑚

𝒓𝒂𝟐 236.060 𝑚𝑚 235.908 𝑚𝑚 235.551 𝑚𝑚

𝒓𝒇𝟏 80.117 𝑚𝑚 80.536 𝑚𝑚 80.180 𝑚𝑚

𝒓𝒇𝟐 251.797 𝑚𝑚 251.587 𝑚𝑚 252.300 𝑚𝑚

𝑵𝟏 25 25 25

𝑵𝟐 68 68 68

𝜶𝒏 29° 28.5° 26°

𝒂𝒘 153.796 𝑚𝑚 153.233 𝑚𝑚 153.233 𝑚𝑚

123

CHAPTER 5

5. VERIFICATION AND DISCUSSION OF THE RESULTS

For the verification of the results, the geometrical and material based constraints are

evaluated for external and internal gear pairs. Evaluations of root clearance, top land

thickness, contact ratio, involute clearance, tiff clearance, contact stress, bending

stress and tooth temperature are verified in this section by comparing the results with

Kissoft commercial gear design and analysis tool.

5.1. Verification of External and Internal Gear Pairs by Using Kissoft

The optimized gear pair which is obtained by Case 6 given in Table 4.8 and the

optimized gear pair which is obtained by Case 11 given in Table 4.20 are verified in

this section for the verification of the external and internal gear pairs, respectively.

The geometrical parameters and material based parameters are evaluated for the

optimized gear pairs. Results are compared with the results of Kissoft commercial gear

design and analysis tool. Results of Case 6 and Kissoft commercial tool are given in

Table 5.1. Tooth contact temperature of the optimized gear pair in Case 6 is obtained

by Kissoft and given in Figure 5-1. Results of the Case 11 and Kissoft commercial

tool are given in Table 5.2. Tooth contact temperature of the optimized gear pair in

Case 11 is obtained by Kissoft and given in Figure 5-2.

124

Table 5.1 Results of the verification case study for external gear pairs

Parameter Current Study Kıssoft

𝑟1 82.478 𝑚𝑚 82.478 𝑚𝑚

𝑟2 325.2 𝑚𝑚 325.2 𝑚𝑚

𝑟𝑏1 75.694 𝑚𝑚 75.694 𝑚𝑚

𝑟𝑏2 298.45 𝑚𝑚 298.45 𝑚𝑚

𝛼𝑤𝑡 23.402° 23.402°

𝑟𝑡𝑓1 87.909 𝑚𝑚 87.909 𝑚𝑚

𝑟𝑡𝑓2 329.688 𝑚𝑚 329.688 𝑚𝑚

𝑅𝑓1 76.995 𝑚𝑚 77.028 𝑚𝑚

𝑅𝑓2 319.171 𝑚𝑚 319.235 𝑚𝑚

𝑃𝑏𝑡 13.589 𝑚𝑚 13.589 𝑚𝑚

𝑆𝑛1𝑏 7.498 𝑚𝑚 7.498 𝑚𝑚

𝑆𝑛2𝑏 6.076 𝑚𝑚 6.076 𝑚𝑚

𝑠𝑛𝑎𝑛1 2.346 𝑚𝑚 2.346 𝑚𝑚

𝑠𝑛𝑎𝑛2 2.152 𝑚𝑚 2.152 𝑚𝑚

𝑍 23.391 𝑚𝑚 23.392 𝑚𝑚

𝑚𝑝 1.721 1.721

𝑚𝐹 2.994 2.994

𝑃𝑥 38.572 𝑚𝑚 38.572 𝑚𝑚

𝑐1 1.414 𝑚𝑚 1.414 𝑚𝑚 𝑐2 1.885 𝑚𝑚 1.885 𝑚𝑚

𝑇𝑖𝑓𝑓1 1.642 𝑚𝑚 1.609 𝑚𝑚

𝑇𝑖𝑓𝑓2 1.472 𝑚𝑚 1.408 𝑚𝑚

𝐼𝑐1 1.301 𝑚𝑚 1.334 𝑚𝑚

𝐼𝑐2 20.721 𝑚𝑚 20.785 𝑚𝑚

𝐽1 0.6728 0.661

𝐽2 0.6827 0.661

𝑠𝑡1 441.836 𝑀𝑃𝑎 459.76 𝑀𝑃𝑎

𝑠𝑡2 435.451 𝑀𝑃𝑎 459.69 𝑀𝑃𝑎

𝐼 0.272 0.272

𝑠𝑐 1072 𝑀𝑃𝑎 1072 𝑀𝑃𝑎

𝜃𝑀 141.08 ℃ 153.53 ℃

125


with angle of rotation

126

Table 5.2 Results of the verification case study for internal gear pairs

Parameter Current Study Kıssoft

𝑟1 89.417 𝑚𝑚 89.417 𝑚𝑚

𝑟2 243.213 𝑚𝑚 243.213 𝑚𝑚

𝑟𝑏1 78.067 𝑚𝑚 78.067 𝑚𝑚

𝑟𝑏2 212.343 𝑚𝑚 212.343 𝑚𝑚

𝛼𝑤𝑡 29.182° 29.182°

𝑟𝑡𝑓1 95.630 𝑚𝑚 95.630 𝑚𝑚

𝑟𝑡𝑓2 236.285 𝑚𝑚 236.285 𝑚𝑚

𝑅𝑓1 81.481 𝑚𝑚 83.159 𝑚𝑚

𝑅𝑓2 251.251 𝑚𝑚 249.092 𝑚𝑚

𝑷𝒃𝒕 19.620 𝑚𝑚 19.620 𝑚𝑚

𝑆𝑛1𝑏 11.0277 𝑚𝑚 11.0277 𝑚𝑚

𝑆𝑛2𝑏 11.0281 𝑚𝑚 11.0281 𝑚𝑚

𝑠𝑛𝑎𝑛1 3.393 𝑚𝑚 3.393 𝑚𝑚

𝑠𝑛𝑎𝑛2 3.396 𝑚𝑚 3.396 𝑚𝑚

𝑍 26.581 𝑚𝑚 26.581 𝑚𝑚

𝑚𝑝 1.355 1.355

𝑚𝐹 0.684 0.684

𝑃𝑥 183.026 𝑚𝑚 183.026 𝑚𝑚

𝑐1 2.147 𝑚𝑚 2.147 𝑚𝑚 𝑐2 2.146 𝑚𝑚 2.146 𝑚𝑚 𝐼𝑐1 3.414 𝑚𝑚 5.092 𝑚𝑚

𝐼𝑐2 23.717 𝑚𝑚 23.717 𝑚𝑚

𝐽1 0.6396 0.681

𝐽2 0.8032 0.853

𝑠𝑡1 352.1888 𝑀𝑃𝑎 332.11 𝑀𝑃𝑎

𝑠𝑡2 280.4582 𝑀𝑃𝑎 265.09 𝑀𝑃𝑎

𝐼 0.3875 0.387

𝑠𝑐 915 𝑀𝑃𝑎 915 𝑀𝑃𝑎

𝜃𝑀 135.6 ℃ 125.0 ℃

127

Figure 5-2 Tooth contact temperature variation of the optimized gear pair in Case

11 with angle of rotation

5.2. Discussion of the Sudden Changes in the Value of the Objective Function

When Figure 4-4, Figure 4-8, Figure 4-11, Figure 4-14 and Figure 4-15 are considered,

it is seen that changes in the optimum center distance values are respectable. The

reason of the sudden changes in the optimum center distance is discussed in this

section. The optimized gear pair obtained by Case 8F given in Table 4.8 is considered

to discuss the sudden changes in the value of the objective function.

When Figure 4-4 is considered, it is seen that the optimum center distance is about

435 𝑚𝑚. However, when the helix angle is 22° and 24° the optimum center distance

is obtained as about 411 𝑚𝑚 and 420 𝑚𝑚 respectively.

It is stated in Table 4.8 that the optimum values of the pinion number of teeth, the gear

number of teeth, the normal module, the pressure angle, the helix angle and the shifting

coefficient are obtained as 37, 145, 4.4 𝑚𝑚, 23°, 23° and 0, respectively. The

optimum center distance is obtained as 434.979 𝑚𝑚.

128

Analysis of Case 21 is conducted around the optimum point of 8F which is given in

Table 4.8 by applying the initial assignments of the design variables given in Table

5.3. However, the optimization eliminations are not implemented in this analysis to

obtain the all possibilities and to observe the reasons of not converging to a more

optimum point.

Table 5.3 Initial assignments of case 21


21 35 ∶ 1 ∶ 40 4.1 ∶ 0.3 ∶ 4.7 20 ∶ 1 ∶ 26 23 −0.2 ∶ 0.2 ∶ 0.2

𝒏𝟏 1500 𝑟𝑝𝑚

𝒏𝟐 381.679 𝑟𝑝𝑚

𝑷𝒐𝒘𝒆𝒓 1700 𝑘𝑊


0.8 0.1 1.20


0.8 0.1 1.20


−1.5 0.1 −1.10


−1.5 0.1 −1.10

As seen from Table 5.3, the given initial assignments of the design variables in Case

21 covers the values of the optimized design variables obtained in Case 8F given in

Table 4.8. 157500 results are obtained without optimization eliminations. The

different center distances among the obtained results are given in Table 5.4.

129

Table 5.4 Center distance variation with normal module, pinion number of teeth and

gear number of teeth in Case 21

𝑚𝑛 𝛽 𝑧1 𝑧2 𝑎𝑤

4.1 𝑚𝑚 23° 35 138 385.278 𝑚𝑚

4.1 𝑚𝑚 23° 37 145 405.321 𝑚𝑚

4.1 𝑚𝑚 23° 38 149 416.456 𝑚𝑚

4.1 𝑚𝑚 23° 40 157 438.727 𝑚𝑚

4.4 𝑚𝑚 23° 35 138 413.469 𝑚𝑚

4.4 𝑚𝑚 23° 37 145 434.979 𝑚𝑚

4.4 𝑚𝑚 23° 38 149 446.929 𝑚𝑚

4.4 𝑚𝑚 23° 40 157 470.829 𝑚𝑚

4.7 𝑚𝑚 23° 35 138 441.660 𝑚𝑚

4.7 𝑚𝑚 23° 37 145 464.636 𝑚𝑚

4.7 𝑚𝑚 23° 38 149 477.401 𝑚𝑚

4.7 𝑚𝑚 23° 40 157 502.931 𝑚𝑚

As given in Table 4.8, the optimized center distance value is 434.979 𝑚𝑚 in Case 8F.

As seen from Table 5.4, the center distance values which is lower than 434.979 𝑚𝑚

are 385.278 𝑚𝑚, 405.321 𝑚𝑚, 413.469 𝑚𝑚 and 416.456 𝑚𝑚. However, the

iteration does not converge to a center distance value which is lower than

434.979 𝑚𝑚 in Case 8F.

Addendum and dedendum radii for each gear pair set are evaluated by using the

algorithms given in Table 3.9, Table 3.10, Table 3.13 and Table 3.14. When Table 5.3

is considered it is seen that the maximum addendum and dedendum radii for each gear

pair set are obtained by using 𝑦2, 𝑦4, 𝑦6 and 𝑦7. The maximum addendum radii are

1.20 𝑚𝑡 higher than the pitch radii of each gear pair set while the maximum dedendum

radii are 1.10 𝑚𝑡 lower than the pitch radii of each gear pair set. The maximum

possible addendum and dedendum radii are given in Table 5.5 and Table 5.6,

respectively.

130

Table 5.5 The maximum addendum radii of the pinion and the gear with normal

module, pinion number of teeth and gear number of teeth in Case 21

𝑚𝑛 𝛽 𝑧1 𝑧2 𝑟𝑎1 𝑟𝑎2 4.1 𝑚𝑚 23° 35 138 83.291 𝑚𝑚 312.676 𝑚𝑚

4.1 𝑚𝑚 23° 37 145 87.745 𝑚𝑚 328.266 𝑚𝑚

4.1 𝑚𝑚 23° 38 149 89.972 𝑚𝑚 337.174 𝑚𝑚

4.4 𝑚𝑚 23° 35 138 89.386 𝑚𝑚 335.555 𝑚𝑚

Table 5.6 The maximum dedendum radii of the pinion and the gear with normal

module, pinion number of teeth and gear number of teeth in Case 21

𝑚𝑛 𝛽 𝑧1 𝑧2 𝑟𝑓1 𝑟𝑓2

4.1 𝑚𝑚 23° 35 138 73.047 𝑚𝑚 302.432 𝑚𝑚

4.1 𝑚𝑚 23° 37 145 77.501 𝑚𝑚 318.021 𝑚𝑚

4.1 𝑚𝑚 23° 38 149 79.728 𝑚𝑚 326.929 𝑚𝑚

4.4 𝑚𝑚 23° 35 138 78.392 𝑚𝑚 324.561 𝑚𝑚

The optimum center distance can be obtained as 385.278 𝑚𝑚. The normal module is

directly 4.1 𝑚𝑚 and the pinion number of teeth is 35 and the gear number of teeth is

138 with the 23° helix angle and with 385.278 𝑚𝑚 center distance as seen from

Table 5.4. The results of Case 21 are filtered by taking the normal module, the pinion

number of teeth, the gear number of teeth as 4.1 𝑚𝑚 , 35 and 138, respectively. The

addendum radii of the pinion and the gear are taken as 83.291 𝑚𝑚 and 312.676 𝑚𝑚,

respectively. The dedendum radii of the pinion and the gear are taken as 73.047 𝑚𝑚

and 302.432 𝑚𝑚, respectively. There are remaining seven gear pairs among the

results of Case 21 when the design parameters are selected as mentioned above. The

bending stress safety factor and the pinion top land thickness relation are given in

Figure 5-3.

131


-0.2 profile shifting coefficient (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 = 35 , 𝑧2 = 138 , 𝛽 = 23° , 𝑟𝑎1 =83.291 𝑚𝑚 , 𝑟𝑎2 = 312.676 𝑚𝑚 , 𝑟𝑓1 = 73.047 𝑚𝑚 , 𝑟𝑓2 = 302.432 𝑚𝑚)

As seen from Figure 5-3, the pinion bending stress safety factor values are all below

1. The top land thickness values of the pinion are also below the minimum required

top land thickness value. The pinion profile shifting coefficient can be increased to 0

by keeping the other parameters as the same to increase the pinion bending stress

safety factor value. The following results given in Figure 5-4 are obtained with the 0

pinion profile shifting coefficient.

-0,4

0,1

0,6

1,1

1,6

2,1

0

0,2

0,4

0,6

0,8

1

1,2

1,4

20 21 22 23 24 25 26

Top

Lan

d T

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ss (

mm

)

Safe

ty F

acto

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Pinion Bending Stress Safety Factor Gear Bending Stress Safety Factor

Pinion Top Land Thickness Minimum Top Land Thickness

132


0 profile shifting coefficient (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 = 35 , 𝑧2 = 138 , 𝛽 = 23° , 𝑥1 =0 , 𝑟𝑎1 = 83.291 𝑚𝑚 , 𝑟𝑎2 = 312.676 𝑚𝑚 , 𝑟𝑓1 = 73.047 𝑚𝑚 , 𝑟𝑓2 = 302.432 𝑚𝑚)

As seen from Figure 5-4, the pinion bending stress safety factor is increased and the

gear bending stress safety factor is decreased. However, when the bending stress

safety factor values are slightly higher than 1 the top land thickness values of the

pinion are still below the minimum required top land thickness value. The gear

bending stress safety factor values are all below the 1 when the profile shifting

coefficient of the pinion is taken as 0.2. Therefore, the iteration can’t converge to the

optimum center distance of 385.278 𝑚𝑚. Because the bending stress safety factor

values can be increased only by increasing the addendum radii. However, the

maximum possible addendum radii are already implemented.



0,50,60,70,80,911,11,21,31,41,51,61,71,81,922,1

0

0,2

0,4

0,6

0,8

1

1,2

20 21 22 23 24 25 26

Top

Lan

d T

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kne

ss (

mm

)

Safe

ty F

acto

r




133

145 with the 23° helix angle and with the 385.278 𝑚𝑚 center distance as seen from



addendum radii of the pinion and the gear are taken as 87.745 𝑚𝑚 and 328.266 𝑚𝑚




bending stress safety factor and the pinion top land thickness relation is given in Figure

5-5.


-0.2 profile shifting coefficient (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 = 37 , 𝑧2 = 145 , 𝛽 = 23°, 𝑟𝑎1 =87.745 𝑚𝑚 , 𝑟𝑎2 = 328.266 𝑚𝑚 , 𝑟𝑓1 = 77.501 𝑚𝑚 , 𝑟𝑓2 = 318.021 𝑚𝑚

As seen from Figure 5-5, the pinion bending stress safety factor values are below 1.

The top land thickness values of the pinion are also below the minimum required top

-0,4

0,1

0,6

1,1

1,6

2,1

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

20 21 22 23 24 25 26

Top

Lan

d T

hic

kne

ss (

mm

)

Safe

ty F

acto

r


Contact Stress Safety Factor Pinion Bending Stress Safety Factor

Gear Bending Stress Safety Factor Pinion Top Land Thickness

Minimum Top Land Thickness

134

land thickness values. The profile shifting coefficient can be increased to 0.2 to

increase the pinion bending stress value and the pinion top land thickness. The

obtained results with 0.2 pinion profile shifting coefficient are given in Figure 5-6.


0.2 profile shifting coefficient (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 = 37 , 𝑧2 = 145 , 𝛽 = 23° , 𝑟𝑎1 =87.745 𝑚𝑚 , 𝑟𝑎2 = 328.266 𝑚𝑚 , 𝑟𝑓1 = 77.501 𝑚𝑚 , 𝑟𝑓2 = 318.021 𝑚𝑚

It is seen from Figure 5-6 that the gear bending stress safety factor value is higher

than 1 when the pressure angle is 25° and 26°. However, the pinion top land thickness

is still below the minimum required top land thickness value. If the pressure angle is

increased the top land thickness value becomes lower. If the pinion profile shifting

coefficient value is increased the tooth thickness of the gear will be lower. Therefore,

the gear bending stress safety factor values become lower than 1. As a result, the

iteration can’t converge to the optimum center distance of 405.321 𝑚𝑚.

20 21 22 23 24 25 26

1,38

1,58

1,78

1,98

2,18

2,38

0

0,2

0,4

0,6

0,8

1

1,2

1,4

Top

Lan

d T

hic

kne

ss (

mm

)

Safe

ty F

acto

r



Gear Bending Stress Safety Factor Pinion Top Land Thickness


135






addendum radii of the pinion and the gear are taken as 89.386 𝑚𝑚 and 335.555 𝑚𝑚,




bending stress safety factor and the pinion top land thickness relation is given in Figure

5-7.


pressure angle with -0.2 profile shifting coefficient (𝑚𝑛 = 4.4 𝑚𝑚 , 𝑧1 = 35 , 𝑧2 =138 , 𝛽 = 23° , 𝑟𝑎1 = 89.386 𝑚𝑚 , 𝑟𝑎2 = 335.555 𝑚𝑚 , 𝑟𝑓1 = 78.392 𝑚𝑚 , 𝑟𝑓2 =

324.561 𝑚𝑚

-1

-0,5

0

0,5

1

1,5

2

2,5

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

20 21 22 23 24 25 26

mm

Safe

ty F

acto

r



Pinion Top Land Thickness Pinion Root Clearance


136

It is seen from Figure 5-7 that all the pinion and all the gear bending stress safety

factor values are almost higher than 1. However, the pinion root clearance is negative.

The analysis results are filtered for 413.469 𝑚𝑚 center distance. All the constraints

are satisfied. However, it is observed that the gear top land thickness criteria is not

satisfied while the other constraints are satisfied. There are remaining sixteen gear

pairs when the all constraints are satisfied except the gear top land thickness

constraint. The safety factor and the top land thickness relation for remaining gear

pairs is given in Figure 5-8. All the addendum radii which are

0.8 𝑚𝑡 , 0.9 𝑚𝑡 , 1.0 𝑚𝑡 ,1.1 𝑚𝑡 and 1.2 𝑚𝑡 higher than the pitch radii of the gear pairs

obtained in Case 21 are considered. All the dedendum radii which are

1.5 𝑚𝑡 , 1.4 𝑚𝑡 , 1.3 𝑚𝑡 ,1.2 𝑚𝑡 and 1.1 𝑚𝑡 lower than the pitch radii of the gear pairs

obtained in Case 21.

Figure 5-8 Variation of safety factor and top land thickness (𝑚𝑛 = 4.4 𝑚𝑚 , 𝑧1 =35 , 𝑧2 = 138 , 𝛽 = 23° , 𝑥1 = 0 , 0.2 , 𝑟𝑎1 = 𝑎𝑙𝑙 , 𝑟𝑎2 = 𝑎𝑙𝑙 , 𝑟𝑓1 = 𝑎𝑙𝑙 , 𝑟𝑓2 = 𝑎𝑙𝑙

0

0,5

1

1,5

2

2,5

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1

1,05

1,1

1,15

1,2

1,25

Top

Lan

d T

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ss (

mm

)

Gear Pair

Safe

ty F

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Gear Bending Stress Safety Factor Gear Top Land Thickness


Maximum Top Land Thickness

137

As seen from Figure 5-8, the contact stress safety factor, the pinion and the gear

bending stress safety factor are above 1. However, the gear top land thickness

constraint is not satisfied in all steps. Therefore, the iteration does not converge to the

413.469 𝑚𝑚 center distance.




Table 5.4. The results of the Case 21 are filtered by taking the normal module, the

pinion number of teeth, the gear number of teeth as 4.1 𝑚𝑚 , 38 and 149, respectively.

The addendum radii of the pinion and the gear are taken as 89.972 𝑚𝑚 and

337.174 𝑚𝑚 respectively. The dedendum radii of the pinion and the gear are taken

as 79.728 𝑚𝑚 and 326.929 𝑚𝑚, respectively. There are remaining seven gear pairs

among the results of Case 21 when the design parameters are selected as mentioned

above. The bending stress safety factor and the pinion top land thickness relation is

given in Figure 5-9.


pressure angle with -0.2 profile shifting coefficient (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 = 38 , 𝑧2 =149 , 𝛽 = 23° , 𝑥1 = 0 , 𝑟𝑎1 = 89.972 𝑚𝑚 , 𝑟𝑎2 = 337.174 𝑚𝑚 , 𝑟𝑓1 =

79.728 𝑚𝑚 , 𝑟𝑓2 = 326.929 𝑚𝑚

-1

-0,5

0

0,5

1

1,5

2

2,5

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

20 21 22 23 24 25 26

mm

Safe

ty F

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Pressure Angle (degrees)



Pinion Root Clearance

138

As seen from Figure 5-9, the pinion top land thickness is below the required minimum

top land thickness value. The pinion profile shifting coefficient must be increased to

higher values. The pinion profile shifting coefficient is increased to 0.2. The following

results given in Figure 5-10 are obtained.

Figure 5-10 Variation of top land thickness and root clearance vs pressure angle with

0.2 profile shifting coefficient (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 = 38 , 𝑧2 = 149 , 𝛽 = 23° , 𝑟𝑎1 =89.972 𝑚𝑚 , 𝑟𝑎2 = 337.174 𝑚𝑚 , 𝑟𝑓1 = 79.728 𝑚𝑚 , 𝑟𝑓2 = 326.929 𝑚𝑚

As seen from Figure 5-10, the gear top land thickness criteria is not satisfied. The

results are filtered to 416.456 𝑚𝑚 center distance. The pinion profile shifting

coefficient are taken as 0 and 0.2. All the pressure angle and the addendum and the

dedendum radii are implemented without any elimination. The obtained analysis

results are filtered to satisfy all the constraints. However, the gear top land thickness

criteria is not satisfied when all the other constraints are satisfied. The following

results given in Figure 5-11 is obtained.

-1

-0,5

0

0,5

1

1,5

2

0

0,5

1

1,5

2

2,5

3

20 21 22 23 24 25 26

Ro

ot

Cle

aran

ce (

mm

)

Top

Lan

d T

hic

kne

ss (

mm

)

Pressure Angle (degrees)

Pinion Top Land Thickness Gear Top Land Thickness

Minimum Top Land Thickness Root Clearance

Minimum Root Clearance Maximum Root Clearance

139

Figure 5-11 Variation of safety factor and top land thickness (𝑚𝑛 = 4.1 𝑚𝑚 , 𝑧1 =38 , 𝑧2 = 149 , 𝛽 = 23° , 𝑥1 = 0, 0.2 , 𝑟𝑎1 = 𝑎𝑙𝑙 , 𝑟𝑎2 = 𝑎𝑙𝑙 , 𝑟𝑓1 = 𝑎𝑙𝑙 , 𝑟𝑓2 = 𝑎𝑙𝑙

As seen from Figure 5-11, all the safety values are above 1. The pinion top land

thickness criteria is also satisfied. However, top land thickness of the gear is below

the minimum required top land thickness value in all analysis. Therefore, the iteration

does not converge to the 416.456 𝑚𝑚 center distance.

5.3. Verification of the Bending Stress Geometry Factor Evaluation

The gear root evaluation is conducted and given in Chapter 2.1.1.6 and Chapter 2.1.1.7

for external and internal gear pairs respectively. The equations given in Chapter

2.1.1.6 are converted into the transverse plane. The gear root equations in transverse

plane are:


𝑏1)

(5.1)

𝑅𝑓1𝑡𝑦 + 𝐴1𝑐𝑜𝑠(𝜃1 − 𝜃𝑥1) − 𝑏1 = 0 (5.2)

0

0,5

1

1,5

2

2,5

0,98

1

1,02

1,04

1,06

1,08

1,1

1,12

1,14

1,16

1,18

1,2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Top

Lan

d T

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mm

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Gear Pair

Contact Stress Safety Pinion Bending Stress Safety

Gear Bending Stress Safety Gear Top Land Thickness


140

𝑎1 + 𝐴1𝑠𝑖𝑛(𝜃1 − 𝜃𝑥1) − 𝑅𝑓1𝑡𝑥 = 0 (5.3)

𝑎12 + 𝑏1

2 − (𝑟𝑓1 + 𝐴1)2

= 0 (5.4)

√(𝑟𝑓1 + 𝐴1)2− 𝑟𝑏1

2 − 𝐴1 − √𝑅𝑓1𝑛2 − 𝑟𝑏1

2 = 0 (5.5)

The bending stress geometry factor evaluation is conducted by using the gear root

equations in the transverse plane for the optimized gear pair obtained by Case 1 given

in Table 4.4. The results of the bending stress geometry factor evaluation of the

optimized gear pair in Case 1 are given in Table 5.7.

Table 5.7 Comparison of the bending stress geometry factor of the optimized gear pair

obtained by Case 1

Parameters Kissoft Current Study

ℎ𝑓1 6.70 𝑚𝑚 6.70 𝑚𝑚

ℎ𝑓2 6.78 𝑚𝑚 6.75 𝑚𝑚

𝑆𝐹1 7.44 𝑚𝑚 7.43 𝑚𝑚

𝑆𝐹2 7.76 𝑚𝑚 7.74 𝑚𝑚

𝛼𝑛𝐿1 29.10° 29.10°

𝛼𝑛𝐿2 24.31° 24.31°

𝐾𝜓 0.95 0.95

𝐶ℎ 1.28 1.28

𝑌1 0.637 0.635

𝑌2 0.641 0.640

𝐾𝑓1 1.578 1.491

𝐾𝑓2 1.584 1.515

𝐶𝜓 1 1

𝑚𝑁 0.59 0.59

𝐴1 1.1516 𝑚𝑚 1.1516 𝑚𝑚

𝐴2 1.1941 𝑚𝑚 1.1942 𝑚𝑚

𝐴1𝑣 0.6845 𝑚𝑚 1.1168 𝑚𝑚

𝐴2𝑣 0.7745 𝑚𝑚 1.1539 𝑚𝑚

𝐽1 0.683 𝑚𝑚 0.721 𝑚𝑚

𝐽2 0.684 𝑚𝑚 0.716 𝑚𝑚

𝑠𝑡1 352 𝑀𝑃𝑎 332 𝑀𝑃𝑎

𝑠𝑡2 351 𝑀𝑃𝑎 334 𝑀𝑃𝑎

141

CHAPTER 6

6. CONCLUSIONS AND FUTURE WORK

The main concern of the current study is the optimization of cylindrical gear pairs.

Normal module, helix angle, pinion number of teeth, gear number of teeth, addendum

radius of the pinion, dedendum radius of the pinion, addendum radius of the gear,

dedendum radius of the gear and pressure angle are design parameters. Root clearance,

top land thickness, contact ratio, involute clearance and tiff clearance are considered

as the geometrical constraints. Evaluation of contact stress, bending stress and

scuffing probability are conducted as material based constraints. From Table 1.1 it is

seen that studies conducted in the literature usually consider a few parameters as

design parameters. However, ten design parameters are optimized by using the

developed methodology by applying five geometrical constraints and three material

based constraints in the current study. In the literature it is seen that nontraditional

optimization techniques are used in the studies. However, it is given by Wu [3] that

there is always the possibility of finding a local optimum in nontraditional

optimization techniques. Additional methods must be implemented to overcome the

possibility of local optimum. However, when the conducted studies in the literature

are considered, it is seen that the optimization problems are usually consisted of a few

design parameters and the number of constraint functions are not high as evaluated in

the current study.

Furthermore, the rounded gear root evaluation is not conducted in the studies.

However, the current study suggests an optimization method with rounded root

evaluations by considering the arbitrary addendum and dedendum radii for the pinion

and the gear. Therefore, using a nontraditional optimization technique can cause a

local optimum solution in such a detailed gear optimization problem. In the current

study, fine sizing method is implemented for optimization. All the possible gear sets

142

for the given range of the design variables are evaluated in fine sizing method. The

constraints are applied for each of the gear pairs. The gear pairs which don’t satisfy

one of the constraints are eliminated. The possibility of the local optimum solution is

eliminated since all the possible solutions are considered for given range of each

design variables.

The most remarkable contribution of the current study is the evaluation of the root

geometry and the bending stress geometry factor of the gear pairs which have rounded

gear root. The gear root evaluation is conducted and given in Chapter 2.1.1.6 and

Chapter 2.1.1.7 for external and internal gear pairs, respectively. The helical gears

which are produced by grinding method have rounded gear root in the normal plane.

However, when Table 5.1 and Table 5.2 are considered, it is seen that the evaluated

bending stress geometry factor differs from Kissoft commercial tool. The gear root

equations given in Chapter 2.1.1.6 are converted into the transverse plane and given

in equations (5.1), (5.2), (5.3), (5.4) and (5.5). Evaluation of the root geometry and the

bending stress geometry factor is evaluated in the transverse plane for the optimized

gear pair obtained by Case1 given in Table 4.4. The obtained results are given in Table

5.7. It is seen from Table 5.7 that the maximum radii of the gear roots (𝐴1 and 𝐴2) are

the same in Kissoft and in the current study. Therefore, it is concluded that Kissoft

evaluates the gear root in the transverse section. However, the helical gears which are

produced by grinding method have rounded gear root in the normal plane.

Since the maximum radii of the gear roots are the same, it is expected that the stress

correction factor for the pinion and for the gear (𝐾𝑓1 and 𝐾𝑓2) must be the same.

However, it is seen that the maximum radii of the virtual gear roots (𝐴1𝑣 and 𝐴2𝑣) are

different in Kissoft results and in the current study. The difference in the radii of the

virtual gear roots comes from that Kissoft uses the methodology outlined in AGMA

908 [13] while the methodology given in Chapter 2.1.2.2.3 is used in the current study.

However, the methodology given in [13] is applicable for the gears which have

trochoidal gear root. Therefore, the methodology given in [13] does not cover the gears

which are produced by grinding method.

143

The outlined methodology in the current study covers the effect of fully rounded gear

roots and the backlash on the bending stress geometry factor evaluation. The

lubrication regime is also considered in the contact stress evaluation. The outlined

methodology differs from the other studies and Kissoft commercial tool by

considering the fully rounded gear root, the lubrication regime and the gears produced

by grinding method.

The proposed methodology in the current study is suitable for the gear pairs produced

by nontraditional manufacturing methods and the optimized design parameters are

nonstandard. The minimum weight consideration is the most important design

constraint in aerospace applications. The minimum weight consideration requires an

optimization methodology and nonstandard design parameters. In the current study,

the minimum center distance optimization is implemented by obtaining the

nonstandard optimized design parameters. There are lots of studies in aerospace

applications to decrease the weight of an air vehicle. The proposed methodology has

a crucial role in aerospace applications regarding the minimum weight considerations.

However, the standard gear pairs which are designed without an optimization

methodology are not the gear pairs which have the minimum center distance and the

minimum weight. Therefore, the standard gear pairs are not suitable for aerospace

applications.

The proposed methodology in the current study covers the optimization of the single

pair of external and internal gear pairs. Optimization of the planetary stages can be

implemented by using the same methodology as a future work. In the literature, there

are some studies which evaluate the bending stress of the asymmetric gears as outlined

in [10]. Therefore, the proposed optimization methodology can be also expanded into

the optimization of asymmetric external and internal gear pairs. Evaluation of the gear

root and the bending stress geometry factor with the 0.250 𝑚𝑚 normal circular

backlash are conducted for different external and internal gear pairs. The obtained

results are given in Table 6.1 and Table 6.2.

144

Table 6.1 Gear root and bending stress geometry factor evaluation for external gear

pairs with 0.250 𝑚𝑚 normal circular backlash

Study 1 2 3 4 5

𝑚𝑛 3.2 𝑚𝑚 4.1 𝑚𝑚 3.2 𝑚𝑚 3.8 𝑚𝑚 3.2 𝑚𝑚

𝛽 13° 17° 21° 21° 7°

𝑥1 0.2 0.2 0.2 0.3 −0.1

𝑧1 37 40 37 43 40

𝑧2 145 157 145 169 157

𝛼𝑛 23° 23° 21.5° 24° 25°

𝑎𝑤 298.86 𝑚𝑚 422.303 𝑚𝑚 311.917 𝑚𝑚 431.457 𝑚𝑚 317.567 𝑚𝑚

𝒓𝒂𝟏 64.698 𝑚𝑚 90.892 𝑚𝑚 67.525 𝑚𝑚 92.397 𝑚𝑚 67.382 𝑚𝑚

𝒓𝒂𝟐 241.058 𝑚𝑚 340.414 𝑚𝑚 251.591 𝑚𝑚 347.201 𝑚𝑚 256.633 𝑚𝑚

𝒓𝒇𝟏 56.816 𝑚𝑚 80.602 𝑚𝑚 58.956 𝑚𝑚 82.628 𝑚𝑚 59.967 𝑚𝑚

𝒓𝒇𝟐 233.176 𝑚𝑚 330.125 𝑚𝑚 243.364 𝑚𝑚 337.839 𝑚𝑚 249.218 𝑚𝑚

𝐴1 1.197 𝑚𝑚 1.435 𝑚𝑚 1.169 𝑚𝑚 0.878 𝑚𝑚 1.284 𝑚𝑚

𝐴2 1.244 𝑚𝑚 1.486 𝑚𝑚 1.283 𝑚𝑚 1.391 𝑚𝑚 1.031 𝑚𝑚

𝐴1𝑣 1.196 𝑚𝑚 1.432 𝑚𝑚 1.166 𝑚𝑚 0.875 𝑚𝑚 1.284 𝑚𝑚

𝐴2𝑣 1.243 𝑚𝑚 1.484 𝑚𝑚 1.281 𝑚𝑚 1.388 𝑚𝑚 1.031 𝑚𝑚

𝐽1 0.677 0.690 0.632 0.694 0.611

𝐽2 0.671 0.685 0.645 0.660 0.691

Table 6.2 Gear root and bending stress geometry factor evaluation for internal gear

pairs with 0.250 𝑚𝑚 normal circular backlash

Study 1 2 3 4 5

𝑚𝑛 7.1 𝑚𝑚 4.7 𝑚𝑚 7.1 𝑚𝑚 4.5 𝑚𝑚 7.1 𝑚𝑚

𝛽 7° 11° 6° 11° 10°

𝑥1 0 −0.1 0 0.2 0

𝑧1 25 25 25 26 25

𝑧2 68 68 68 71 68

𝛼𝑛 29° 28° 29° 26° 29°

𝑎𝑤 153.796 𝑚𝑚 102.941 𝑚𝑚 153.491 𝑚𝑚 103.145 𝑚𝑚 155.005 𝑚𝑚

𝒓𝒂𝟏 95.854 𝑚𝑚 63.680 𝑚𝑚 95.664 𝑚𝑚 64.638 𝑚𝑚 96.608 𝑚𝑚

𝒓𝒂𝟐 236.060 𝑚𝑚 157.524 𝑚𝑚 235.591 𝑚𝑚 158.614 𝑚𝑚 237.915 𝑚𝑚

𝒓𝒇𝟏 80.117 𝑚𝑚 53.146 𝑚𝑚 79.958 𝑚𝑚 53.635 𝑚𝑚 80.747 𝑚𝑚

𝒓𝒇𝟐 251.797 𝑚𝑚 168.058 𝑚𝑚 251.297 𝑚𝑚 169.158 𝑚𝑚 253.775 𝑚𝑚

𝐴1 1.946 𝑚𝑚 1.570 𝑚𝑚 1.959 𝑚𝑚 1.178 𝑚𝑚 1.890 𝑚𝑚

𝐴2 1.174 𝑚𝑚 1.012 𝑚𝑚 1.191 𝑚𝑚 1.203 𝑚𝑚 1.107 𝑚𝑚

𝐴1𝑣 1.944 𝑚𝑚 1.566 𝑚𝑚 1.958 𝑚𝑚 1.175 𝑚𝑚 1.887 𝑚𝑚

𝐴2𝑣 1.174 𝑚𝑚 1.012 𝑚𝑚 1.191 𝑚𝑚 1.203 𝑚𝑚 1.107 𝑚𝑚

𝐽1 0.640 0.879 0.625 0.914 0.679

𝐽2 0.803 1.152 0.786 1.116 0.850

145

REFERENCES

[1] Y. M. Mohan and T. Seshaiah, “ Spur Gear Optimization by Using Genetic

Algorithm, ’’ International Journal of Engineering Research and Applications,

vol. 2, No. 1, pp. 311-318, Jan-Feb.2012.

[2] A. K. Singh, H. P. Gangwar, R. Saxena and A. Misra, “ Optimization of Internal

Spur Gear Design Using Genetic Algorithm, ’’ MIT International Journal of

Mechenical Engineering, vol. 2, No. 1, pp. 22-30, Jan.2012.

[3] R. Wu, “Gear Weight Optimization Design Model Based on Disturbance and

Simulated Annealing Algorithm, ’’Metallurgical and Mining Industry, vol. 51,

No. 8, pp. 502-507, 2015.

[4] V. Revar, D. Parmar and H. Shah, “Design, Modelling and Stress Analysis of

High Speed Helical Gear on Basis of Bending Strength and Contact Strength by

Changing Face Width and Helix Angle, ’’International Journal For Technological

Research In Engineering, vol.3, Issue 9, pp. 1975-1980, May 2016.

[5] A. L. Kapelevich and Y. V. Shekhtman, “Direct Gear Design: Bending Stress

Minimization, ’’Gear Technology, vol.20, No. 5, pp. 44-47 , September-October

2003.

[6] S. Padmanabhan, S. Ganesan, M. Chandrasekaran, and V. Srinivasa Raman,

“Gear pair design optimization by genetic algorithm and FEA,” Proc. Int. Conf.

Front. Automob. Mech. Eng. - 2010, FAME-2010, No. November, pp. 301–307,

2010.

146

[7] Ram Gopal, Rajiv Suman and R.S. Jadoun, “ Optimization of Helical Gear

Design Using Genetic Algorithm for Center Distance, ’’International Journal of

Advance and Innovation, vol.4, Issue 3, pp. 543-548, 2016.

[8] Dr. Rajiv Suman, Ram Gopal, Anshika Gupta and Praveen Kumar Singh “ Use

of Genetic Algorithm for Center Distance to Optimize the Helical Gear Design,

’’Seminar on Advances in Technology to Mitigate the Effect of Natural Hazards,

organized by The Institution of Engineers (India), Pantnagar Local Centre, March

2-3, 2017.

[9] Paridhi Rai, Asim Gopal Barman, “Design Optimization of Spur Gear Using

SA and RCGA, ’’Journal of the Brazilian Society of Mechanical Sciences and

Engineering, vol.40, Issue 5, pp. 1-8, 2018.

[10] Mahir Gökhan Orak, “Investigation of Asymmetric Gear Tooth Bending Stress

Formulation, ’’Middle East Technical University, March 2018.

[11] International Standard Organization, ‘‘Gears – Cylindrical involute gears and

gear pairs – Concepts and geometry ISO 21771,’’ vol.01, no. September, p.31,

2007.

[12] American Gear Manufacturers Association, “Method for Specifying the

Geometry of Spur and Helical Gears AGMA 913-A98,” vol. 98, no. March, p. 4-

8,12, 1998.

[13] American Gear Manufacturers Association, “Geometry Factors for

Determining the Pitting Resistance and Bending Strength of Spur, Helical and

Herringbone Gear Teeth AGMA 908-B89,” vol. 89, no. August, p. 5-8,16 ,1989.

147

[14] American Gear Manufacturers Association, “Fundamental Rating Factors and

Calculation Methods for Involute Spur and Helical Gear Teeth AGMA 2001-

D04,” vol. 04, no. December, p. 9-18, 20-24,33-42, 2004.

[15] Engineering Sciences Data Unit, ‘‘The design of spur and helical involute

gears. A procedure compatible with BS 436: Part 3: method for calculation of

contact and root bending stress limitations for metallic involute gears ESDU

88033,’’ vol.3, no. June, p.32, 16, 2012

[16] American Gear Manufacturers Association, “Effect of Lubrication on Gear

Surface Distress AGMA 925-A03,” vol. 03, no. January, p. 7-25, 2013.

[17] Stephen P.Radzevich, Dudley’s Handbook of Practical Gear Design and

Manufacture, ’’ CRC Press, Second Edition, pp. 311-318, 2012.

[18] International Organization for Standardization, ‘‘Calculation of Load Capacity

of Spur and Helical Gears ISO/DIS 6336-6.’’ no. September, p.15,16, 2005

149

7. APPENDICES

A. Overload factor, 𝑲𝒐

Overload factor,𝐾𝑜, is given in [18]. Examples for driving machines with various

characteristics are given in Table A.1.

Table A.1 Examples for driving machines with various working characteristics [18]

In aerospace industry, gas turbines are generally used. Therefore, driving machine

characteristic of the driving machine is specified as light shocks.

150

Table A.2 Examples of working characteristics of driven machine [18]

Since the gear pairs are considered in this study, working characteristic of the driven

machine is specified as uniform.

After determining the characteristics of the driving and driving machine, overload

factor is specified by using Table A.3

Table A.3 Overload factor, 𝐾𝑜 [18]

As seen from Table A.3, overload factor is ′1.10′ for light shock driving machine and

uniform driven machine.

151

B. Load Distribution Factor, 𝑲𝒎

In [14], the load distribution factor (𝐾𝑚) is given as:

𝐾𝑚 = 𝐶𝑚𝑓 (B.1)

For relatively stiff gear designs having gears mounted between bearings and relatively

free from externally caused deflections, the following approximate method may be

used:

𝐶𝑚𝑓 = 1.0 + 𝐶𝑚𝑐(𝐶𝑝𝑓𝐶𝑝𝑚 + 𝐶𝑚𝑎𝐶𝑒) (B.2)

Lead correction factor, 𝐶𝑚𝑐, is 0.8 for gear with leads properly modified by crowning

or lead correction.

The values for 𝐶𝑝𝑓 can be determined by the following equations:

when 𝐹 ≤ 1.0

𝐶𝑝𝑓 =𝐹

10𝑑𝑤1− 0.025

(B.3)

when 1.0 < 𝐹 ≤ 17

𝐶𝑝𝑓 =𝐹

10𝑑𝑤1− 0.0375 + 0.0125𝐹

(B.4)

when 17 < 𝐹 ≤ 40

𝐶𝑝𝑓 =𝐹

10𝑑𝑤1− 0.1109 + 0.0207𝐹 − 0.000228𝐹2

(B.5)

For values of (𝐹 ⁄ (10𝑑_𝑤1 )) less than 0.05, 0.05 is used for 𝐶𝑝𝑓.

𝐶𝑝𝑚 is 1.0 for straddle mounted pinions with 𝑆1/𝑆 < 0.175 and 𝐶𝑝𝑚 is 1.1 for straddle

mounted pinions with 𝑆1/𝑆 ≥ 0.175.

152

Fig. B.1 Evaluation of 𝑆 and 𝑆1 [14]

𝐶𝑝𝑚 is taken as unity in this study.

The mesh alignment factor, 𝐶𝑚𝑎, is given as:

𝐶𝑚𝑎 = 𝐴 + 𝐵(𝐹) + 𝐶(𝐹)2 (B.6)

Table B.1 Empirical constants; 𝐴, 𝐵 𝑎𝑛𝑑 𝐶 [14]

Extra precision enclosed gear units are considered in this study.

𝐶𝑒 is 0.80 when the gearing is adjusted at assembly;

is 0.80 when the compatibility of the gearing is improved by lapping;

is 1.0 for all other conditions.

Super finished gears are considered in this study. Therefore, 𝐶𝑒 is taken as 0.8.

153

C. Rim Thickness Factor

Rim thickness factor is given in Fig. A.2.

Fig. C.1 Rim thickness factor, 𝐾𝐵 [14]

In this study, 𝒎𝑩 is considered as higher than 1.2. Therefore, rim thickness factor

is taken as unity.

154

D. Other Factors

According to [14], where gearing is manufactured using process controls which

provide tooth accuracies which correspond to ‘very accurate gearing ‘, or where the

design and manufacturing techniques ensure a low transmission error which is

equivalent to this accuracy, values of 𝐾𝑣 between 1.02 and 1.11 may be used,

depending on the specifier’s experience with similar applications and the degree of

accuracy actually achieved. Since aerospace gears have a very high level of accuracy,

dynamic factor is taken as 1.02.

The size factor, 𝐾𝑠 is related with non-uniformity of material properties. Since the gear

materials used in the current study are Grade 3 quality, the size factor is taken as unity.

The surface condition factor, 𝐶𝑓 is related with surface finish and residual stresses.

Since the gear teeth are grinded from solid and shot peened, the tensile residual

stresses on the gear teeth are avoided even the compressive residual stresses arise.

Also since the super finish is applied on the gear teeth, the surface roughness is

minimized. Therefore, 𝐶𝑓 is taken as unity.

155

E. Scuffing Risk Evaluation

Scuffing risk evaluation is given in [16]. For random variables that follow normal

(Gaussian) distributions, the following procedure can be used to calculate probabilities

of failure in the range of 5 % to 95 %.

𝑥 =(𝑦 − 𝜇𝑦)

𝜎𝑦

(E.1)

𝑍𝑄 = 0.3989422804𝑒[−0.5(𝑥)2] (E.2)

Table E.1 Constant parameters for Scuffing Evaluation [16]

Parameter Value

𝒃𝟏 0.319381530

𝒃𝟐 −0.356563782

𝒃𝟑 1.781477937

𝒃𝟒 −1.821255978

𝒃𝟓 1.330274429

𝒑 0.2316419

𝑡 =1

1 + 𝑝|𝑥|

(E.3)

Evaluation of 𝑄

if |𝑥| > 1.6448, then:

𝑄 = 0.05;

(E.4)

156

else

𝑄 = 𝑍𝑄(𝑏1𝑡 + 𝑏2𝑡2 + 𝑏3𝑡

3 + 𝑏4𝑡4 + 𝑏5𝑡

5)

(E.5)

Probability of failure

if 𝑥 > 0, then:

𝑝𝑟𝑜𝑏𝑎𝑏𝑙𝑖𝑡𝑦 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 = 1 − 𝑄

(E.6)

else

𝑝𝑟𝑜𝑏𝑎𝑏𝑙𝑖𝑡𝑦 𝑜𝑓 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 = 𝑄

(E.7)

157

F. Evaluation of Material Parameter

Material parameter is given in [16]:

𝐺 = 𝛼𝐸𝑟

(F.1)

Where

𝛼 = 𝑘(𝜂𝑀)𝑠 (F.2)

𝛼 = 𝑘(𝜂𝑀)𝑠 (F.3)

𝜂𝑀 = 10𝑔 − 0.9 (F.4)

𝑔 = 10𝑐(𝜃𝑀 + 273.15)𝑑 (F.5)

𝑐 = log10[log10(𝜂40 + 0.9)] − 2.495752𝑑 (F.6)

𝑑 = 13.13525 log10 [log10(𝜂100 + 0.9)

log10(𝜂40 + 0.9)]

(F.7)

Viscosity and pressure-viscosity coefficient values for different lubricants are given

in Table F.1. MIL – L – 23699E is considered in this study.

Dynamic viscosity of the oil at 40 ℃ is:

𝜂40 = 22.56448 𝑚𝑃𝑎. 𝑠 (F.8)

Dynamic viscosity of the oil at 100 ℃ is:

𝜂100 = 4.591235 𝑚𝑃𝑎. 𝑠 (F.9)

𝑘 = 0.006515 (F.10)

158

Table F.1 Data for determining viscosity and pressure-viscosity coefficient [16]

159

G. Backlash and Tip Round Values

When the backlash is required, the following values given in Table G.1 can be used.

For tip chamfer, the following values given in Table G.2 can be used.

Table G.1 Recommended backlash values [17]

Table G.2 Tip round values [17]

160

H. Optimization Flow Chart

START

𝐾𝑜 , 𝐾𝑣 , 𝐾𝑠 , 𝐾𝑚 , 𝐶𝑓 , 𝐾𝐵 , 𝑛1 , 𝑛2

𝑃 , 𝑣1 , 𝑣2 , 𝐸1 , 𝐸2 , 𝑅𝑎1𝑥 , 𝑅𝑎2𝑥

, 𝜆𝑀1 , 𝜌𝑀1 ,

𝑐𝑀1 , 𝜆𝑀2 , 𝜌𝑀2 , 𝑐𝑀2 , 𝐾 , 𝐵𝑀1 , 𝐵𝑀2 , 𝜃𝑜𝑖𝑙 , 𝑘𝑠𝑢𝑚𝑝 , 𝜂100 , 𝜂40 , 𝑘 , 𝑠 , 𝑏1 , 𝑏2 , 𝑏3 , 𝑏4 , 𝑏5 , 𝑝

𝑚𝐺 , 𝑠𝑎𝑐 , 𝑠𝑎𝑡 , 𝑚𝐺

Initial Assignments of the

Design Variables

Table 3.7

Cross combination of

𝑚𝑛 𝛼𝑛 𝛽 𝑥1 [𝑧1 𝑧2] Table 3.8

Evaluation of 𝑟𝑎1

Table 3.9

Evaluation of 𝑟𝑓2

Table 3.10

Evaluation of 𝑐2

Eq. (2.2) (2.4)

Gear Root Clearance

Elimination Table 3.11

Pinion Top Land Thickness

Evaluation

Chapter 2.1.1.2 / 2.1.1.3

Pinion Top Land Thickness

Elimination

Table 3.12

Evaluation of 𝑟𝑎2

Table 3.13

Evaluation of 𝑟𝑓1

Table 3.14

Combination of

𝑚𝑛 𝛼𝑛 𝛽 𝑥1 [𝑧1 𝑧2] 𝑟𝑎1 𝑟𝑓2

Eq. (3.17) - (3.25)

𝑃1

161

Evaluation of 𝑐1

Eq. (2.1) (2.3)

Pinion Root Clearance

Elimination

Table 3.15

Gear Top Land Thickness

Evaluation

Chapter 2.1.1.2 / 2.1.1.3

Gear Top Land Thickness

Elimination

Table 3.16

Evaluation of 𝑚𝑐

Chapter 2.1.1.4 / 2.1.1.5

Contact Ratio Elimination

Table 3.17

Evaluation of 𝐼𝐶1

Eq. (2.110)

Pinion Involute Clearance

Elimination

Table 3.18

Combination of

𝑚𝑛 𝛼𝑛 𝛽 𝑥1 [𝑧1 𝑧2] 𝑟𝑎1 𝑟𝑓2 𝑟𝑎2 𝑟𝑓1

Eq. (3.31) - (3.38) Evaluation of 𝐼𝐶2

Eq. (2.111) (2.112)

Gear Involute Clearance

Elimination

Table 3.19

Evaluation of 𝑇𝑖𝑓𝑓1

Eq. (2.113)

Pinion Tiff Clearance

Elimination

Table 3.20

Evaluation of 𝑇𝑖𝑓𝑓2

Eq. (2.114) (2.115)

Gear Tiff Clearance Elimination

Table 3.21

Contact Stress Evaluation

Chapter 2.1.2.1

𝑃1

𝑃2

162

Virtual Spur Gear Evaluation

Chapter 2.1.2.2.1

Load Angle and Load

Radius Evaluation

Chapter 2.1.2.2.2 /

2.1.2.2.6

Virtual Gear Root Evaluation

Chapter 2.1.2.2.3 / 2.1.2.2.7

Critical Section Determination

Chapter 2.1.2.2.4 / 2.1.2.2.8

Critical Radius Evaluation

Chapter 2.1.2.2.5 / 2.1.2.2.9

𝑃2

Bending Stress Evaluation

Chapter 2.1.2.2

Bending Stress Elimination

Table 3.23 / Table 3.24

Contact Stress Elimination

Table 3.22

Profile Radii of Curvature

Chapter 2.1.2.3.1

Gear Tooth Velocities and

Loads

Chapter 2.1.2.3.2

Load Sharing Factor

Chapter 2.1.2.3.3

Evaluation of Maximum

Flash Temperature

Chapter 2.1.2.3.4

Scuffing Evaluation

Chapter 2.1.2.3

Scuffing Elimination

Table 3.25

Minimum Center Distance

Optimum Gear Pair

GEAR TOOTH OPTIMIZATION FOR NONSTANDARD …

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